Excitation and use of guided surface wave modes on lossy media

ABSTRACT

Disclosed are various embodiments systems and methods for transmission and reception of electrical energy along a surface of a terrestrial medium. A polyphase waveguide probe that transmits electrical energy in the form of a guided surface wave along a surface of a terrestrial medium. A receive circuit is used to receive the electrical energy.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to co-pending U.S. patent application Ser. No. 13/789,538 filed on Mar. 7, 2013 and entitled “EXCITATION AND USE OF GUIDED SURFACE WAVE MODES ON LOSSY MEDIA.”

BACKGROUND

For over a century, signals transmitted by radio waves involved radiation fields launched using conventional antenna structures. In contrast to radio science, electrical power distribution systems in the last century involved the transmission of energy guided along electrical conductors. This understanding of the distinction between radio frequency (RF) and power transmission has existed since the early 1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views.

FIG. 1 is a chart that depicts a field strength as a function of distance for a guided electromagnetic field and a radiated electromagnetic field.

FIG. 2 is a drawing that illustrates a propagation interface with two regions employed for transmission of a guided surface wave according to various embodiments of the present disclosure.

FIG. 3 is a drawing that illustrates a polyphase waveguide probe disposed with respect to a propagation interface of FIG. 2 according to an embodiment of the present disclosure.

FIG. 4 is a drawing that provides one example illustration of a phase shift in a ground current that facilitates the launching of a guided surface-waveguide mode on a lossy conducting medium in the propagation interface of FIG. 3 according to an embodiment of the present disclosure.

FIG. 5 is a drawing that illustrates a complex angle of insertion of an electric field synthesized by the polyphase waveguide probes according to the various embodiments of the present disclosure.

FIG. 6 is a schematic of a polyphase waveguide probe according to an embodiment of the present disclosure.

FIGS. 7A-J are schematics of specific examples of the polyphase waveguide probe of FIG. 6 according to various embodiments of the present disclosure.

FIGS. 8A-C are graphs that illustrate field strengths of guided surface waves at select transmission frequencies generated by the various embodiments of polyphase waveguide probes according to the various embodiments of the present disclosure.

FIG. 9 shows one example of a graph of experimental measurements of field strength of a guided surface wave at 59 Megahertz as a function of distance generated by a polyphase waveguide probe according to an embodiment of the present disclosure.

FIG. 10 shows a graph of experimental measurements of the phase as a function of distance of the guided surface wave of FIG. 9 according to an embodiment of the present disclosure.

FIG. 11 shows another example of a graph of experimental measurements of field strength as a function of distance of a guided surface wave generated by a polyphase waveguide probe at 1.85 Megahertz according to an embodiment of the present disclosure.

FIGS. 12A-B depict examples of receivers that may be employed to receive energy transmitted in the form of a guided surface wave launched by a polyphase waveguide probe according to the various embodiments of the present disclosure.

FIG. 13 depicts an example of an additional receiver that may be employed to receive energy transmitted in the form of a guided surface wave launched by a polyphase waveguide probe according to the various embodiments of the present disclosure.

FIG. 14A depicts a schematic diagram representing the Thevenin-equivalent of the receivers depicted in FIGS. 12A-B according to an embodiment of the present disclosure.

FIG. 14B depicts a schematic diagram representing the Norton-equivalent of the receiver depicted in FIG. 13 according to an embodiment of the present disclosure.

DETAILED DESCRIPTION

Referring to FIG. 1, to begin, some terminology shall be established to provide clarity in the discussion of concepts to follow. First, as contemplated herein, a formal distinction is drawn between radiated electromagnetic fields and guided electromagnetic fields.

As contemplated herein, a radiated electromagnetic field comprises electromagnetic energy that is emitted from a source structure in the form of waves that are not bound to a waveguide. For example, a radiated electromagnetic field is generally a field that leaves an electric structure such as an antenna and propagates through the atmosphere or other medium and is not bound to any waveguide structure. Once radiated electromagnetic waves leave an electric structure such as an antenna, they continue to propagate in the medium of propagation (such as air) independent of their source until they dissipate regardless of whether the source continues to operate. Once electromagnetic waves are radiated, they are not recoverable unless intercepted, and, if not intercepted, the energy inherent in radiated electromagnetic waves is lost forever. Electrical structures such as antennas are designed to radiate electromagnetic fields by maximizing the ratio of the radiation resistance to the structure loss resistance. Radiated energy spreads out in space and is lost regardless of whether a receiver is present. The energy density of radiated fields is a function of distance due to geometrical spreading. Accordingly, the term “radiate” in all its forms as used herein refers to this form of electromagnetic propagation.

A guided electromagnetic field is a propagating electromagnetic wave whose energy is concentrated within or near boundaries between media having different electromagnetic properties. In this sense, a guided electromagnetic field is one that is bound to a waveguide and may be characterized as being conveyed by the current flowing in the waveguide. If there is no load to receive and/or dissipate the energy conveyed in a guided electromagnetic wave, then no energy is lost except for that dissipated in the conductivity of the guiding medium. Stated another way, if there is no load for a guided electromagnetic wave, then no energy is consumed. Thus, a generator or other source generating a guided electromagnetic field does not deliver real power unless a resistive load is present. To this end, such a generator or other source essentially runs idle until a load is presented. This is akin to running a generator to generate a 60 Hertz electromagnetic wave that is transmitted over power lines where there is no electrical load. It should be noted that a guided electromagnetic field or wave is the equivalent to what is termed a “transmission line mode.” This contrasts with radiated electromagnetic waves in which real power is supplied at all times in order to generate radiated waves. Unlike radiated electromagnetic waves, guided electromagnetic energy does not continue to propagate along a finite length waveguide after the energy source is turned off. Accordingly, the term “guide” in all its forms as used herein refers to this transmission mode of electromagnetic propagation.

To further illustrate the distinction between radiated and guided electromagnetic fields, reference is made to FIG. 1 that depicts graph 100 of field strength in decibels (dB) above an arbitrary reference in volts per meter as a function of distance in kilometers on a log-dB plot. The graph 100 of FIG. 1 depicts a guided field strength curve 103 that shows the field strength of a guided electromagnetic field as a function of distance. This guided field strength curve 103 is essentially the same as a transmission line mode. Also, the graph 100 of FIG. 1 depicts a radiated field strength curve 106 that shows the field strength of a radiated electromagnetic field as a function of distance.

Of interest are the shapes of the curves 103/106 for radiation and for guided wave propagation. The radiated field strength curve 106 falls off geometrically (1/d, where d is distance) and is a straight line on the log-log scale. The guided field strength curve 103, on the other hand, has the characteristic exponential decay of e^(−αd)/√{square root over (d)} and exhibits a distinctive knee 109. Thus, as shown, the field strength of a guided electromagnetic field falls off at a rate of e^(−αd)/√{square root over (d)}, whereas the field strength of a radiated electromagnetic field falls off at a rate of 1/d, where d is the distance. Due to the fact that the guided field strength curve 103 falls off exponentially, the guided field strength curve 103 features the knee 109 as mentioned above. The guided field strength curve 103 and the radiated field strength curve 106 intersect at a crossover point 113 which occurs at a crossover distance. At distances less than the crossover distance, the field strength of a guided electromagnetic field is significantly greater at most locations than the field strength of a radiated electromagnetic field. At distances greater than the crossover distance, the opposite is true. Thus, the guided and radiated field strength curves 103 and 106 further illustrate the fundamental propagation difference between guided and radiated electromagnetic fields. For an informal discussion of the difference between guided and radiated electromagnetic fields, reference is made to Milligan, T., Modern Antenna Design, McGraw-Hill, 1^(st) Edition, 1985, pp. 8-9, which is incorporated herein by reference in its entirety.

The distinction between radiated and guided electromagnetic waves, made above, is readily expressed formally and placed on a rigorous basis. That two such diverse solutions could emerge from one and the same linear partial differential equation, the wave equation, analytically follows from the boundary conditions imposed on the problem. The Green function for the wave equation, itself, contains the distinction between the nature of radiation and guided waves.

In empty space, the wave equation is a differential operator whose eigenfunctions possess a continuous spectrum of eigenvalues on the complex wave-number plane. This transverse electro-magnetic (TEM) field is called the radiation field, and those propagating fields are called “Hertzian waves”. However, in the presence of a conducting boundary, the wave equation plus boundary conditions mathematically lead to a spectral representation of wave-numbers composed of a continuous spectrum plus a sum of discrete spectra. To this end, reference is made to Sommerfeld, A., “Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie,” Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A., “Problems of Radio,” published as Chapter 6 in Partial Differential Equations in Physics—Lectures on Theoretical Physics: Volume VI, Academic Press, 1949, pp. 236-289, 295-296; Collin, R. E., “Hertzian Dipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20^(th) Century Controversies,” IEEE Antennas and Propagation Magazine, Vol. 46, No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P. F, Krauss, H. L., and Skalnik, J. G., Microwave Theory and Techniques, Van Nostrand, 1953, pp. 291-293, each of these references being incorporated herein by reference in their entirety.

To summarize the above, first, the continuous part of the wave-number eigenvalue spectrum, corresponding to branch-cut integrals, produces the radiation field, and second, the discrete spectra, and corresponding residue sum arising from the poles enclosed by the contour of integration, result in non-TEM traveling surface waves that are exponentially damped in the direction transverse to the propagation. Such surface waves are guided transmission line modes. For further explanation, reference is made to Friedman, B., Principles and Techniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286, 290, 298-300.

In free space, antennas excite the continuum eigenvalues of the wave equation, which is a radiation field, where the outwardly propagating RF energy with E_(z) and H_(φ) in-phase is lost forever. On the other hand, waveguide probes excite discrete eigenvalues, which results in transmission line propagation. See Collin, R. E., Field Theory of Guided Waves, McGraw-Hill, 1960, pp. 453, 474-477. While such theoretical analyses have held out the hypothetical possibility of launching open surface guided waves over planar or spherical surfaces of lossy, homogeneous media, for more than a century no known structures in the engineering arts have existed for accomplishing this with any practical efficiency. Unfortunately, since it emerged in the early 1900's, the theoretical analysis set forth above has essentially remained a theory and there have been no known structures for practically accomplishing the launching of open surface guided waves over planar or spherical surfaces of lossy, homogeneous media.

According to the various embodiments of the present disclosure, various polyphase waveguide probes are described that are configured to excite radial surface currents having resultant fields that synthesize the form of surface-waveguide modes along the surface of a lossy conducting medium. Such guided electromagnetic fields are substantially mode-matched in magnitude and phase to a guided surface wave mode on the surface of the lossy conducting medium. Such a guided surface wave mode may also be termed a Zenneck surface wave mode. By virtue of the fact that the resultant fields excited by the polyphase waveguide probes described herein are substantially mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium, a guided electromagnetic field in the form of a Zenneck surface wave is launched along the surface of the lossy conducting medium. According to one embodiment, the lossy conducting medium comprises a terrestrial medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides for an examination of the boundary value solution to Maxwell's equations derived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck, J., “On the Propagation of Plane Electromagnetic Waves Along a Flat Conducting Surface and their Relation to Wireless Telegraphy,” Annalen der Physik, Serial 4, Vol. 23, Sep. 20, 1907, pp. 846-866. FIG. 2 depicts cylindrical coordinates for radially propagating waves along the interface between a lossy conducting medium specified as Region 1 and an insulator specified as Region 2. Region 1 may comprise, for example, any lossy conducting medium. In one example, such a lossy conducting medium may comprise a terrestrial medium such as the Earth or other medium. Region 2 is a second medium that shares a boundary interface with Region 1 and has different constitutive parameters relative to Region 1. Region 2 may comprise, for example, any insulator such as the atmosphere or other medium. The reflection coefficient for such a boundary interface goes to zero only for incidence at a complex Brewster angle. See Stratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941, p. 516.

According to various embodiments, the present disclosure sets forth various polyphase waveguide probes that generate electromagnetic fields that are substantially mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium comprising Region 1. According to various embodiments, such electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium that results in zero reflection.

To explain further, in Region 2, where e^(jωt) field variation is assumed and where ρ≠0 and z≧0 (z is a vertical coordinate normal to the surface of Region 1, ρ is the radial dimension in cylindrical coordinates), Zenneck's closed-form exact solution of Maxwell's equations satisfying the boundary conditions along the interface are expressed by the following electric field and magnetic field components:

$\begin{matrix} {{H_{2\phi} = {A\;{\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},} & (1) \\ {{E_{2\rho} = {{A\left( \frac{u_{2}}{{j\omega ɛ}_{o}} \right)}{\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;{\gamma\rho}} \right)}}},{and}} & (2) \\ {{E_{2z} = {{A\left( \frac{- \gamma}{{\omega ɛ}_{o}} \right)}{\mathbb{e}}^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\;{\gamma\rho}} \right)}.}}}\;} & (3) \end{matrix}$

In Region 1, where e^(jωt) field variation is assumed and where ρ≠0 and z≦0, Zenneck's closed-form exact solution of Maxwell's equations satisfying the boundary conditions along the interface are expressed by the following electric field and magnetic field components:

$\begin{matrix} {{H_{1\phi} = {A\;{\mathbb{e}}^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},} & (4) \\ {{E_{1\rho} = {{A\left( \frac{- u_{1}}{\sigma_{1} + {j\omega ɛ}_{1}} \right)}{\mathbb{e}}^{u_{1}z}{H_{1}^{(2)}\left( {- {j\gamma\rho}} \right)}}},{and}} & (5) \\ {{E_{1z} = {{A\left( {- \frac{j\;\gamma}{\sigma_{1} + {j\omega ɛ}_{1}}} \right)}{\mathbb{e}}^{u_{1}z}{{H_{0}^{(2)}\left( {{- j}\;{\gamma\rho}} \right)}.}}}\;} & (6) \end{matrix}$ In these expressions, H_(n) ⁽²⁾(−jγρ) is a complex argument Hankel function of the second kind and order n, u₁ is the propagation constant in the positive vertical direction in Region 1, u₂ is the propagation constant in the vertical direction in Region 2, σ₁ is the conductivity of Region 1, ω is equal to 2πf, where f is a frequency of excitation, ∈_(o) is the permittivity of free space, ∈₁ is the permittivity of Region 1, A is a source constant imposed by the source, z is a vertical coordinate normal to the surface of Region 1, γ is a surface wave radial propagation constant, and ρ is the radial coordinate.

The propagation constants in the ±z directions are determined by separating the wave equation above and below the interface between Regions 1 and 2, and imposing the boundary conditions. This exercise gives, in Region 2,

$\begin{matrix} {{u_{2} = \frac{{- j}\; k_{o}}{\sqrt{1 + \left( {ɛ_{r} - {j\; x}} \right)}}},} & (7) \end{matrix}$ and gives, in Region 1, u ₁ =−u ₂(∈_(r) −jx).  (8) The radial propagation constant γ is given by γ=j√{square root over (k _(o) ² +u ₂ ²)},  (9) which is a complex expression. In all of the above Equations,

$\begin{matrix} {{x = \frac{\sigma_{1}}{{\omega ɛ}_{0}}},{and}} & (10) \\ {{k_{0} = {\omega\sqrt{\mu_{o}ɛ_{o}}}},} & (11) \end{matrix}$ where μ_(o) comprises the permeability of free space, ∈_(r) comprises relative permittivity of Region 1. Thus, the surface wave generated propagates parallel to the interface and exponentially decays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) may be considered to be a cylindrically-symmetric, radially-propagating waveguide mode. See Barlow, H. M., and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 10-12, 29-33. The present disclosure details structures that excite this “open boundary” waveguide mode. Specifically, according to various embodiments, a polyphase waveguide probe is provided with charge terminals of appropriate size that are positioned relative to each other and are fed with voltages and/or currents so as to excite the relative phasing of the fields of the surface waveguide mode that is to be launched along the boundary interface between Region 2 and Region 1.

To continue further, the Leontovich impedance boundary condition between Region 1 and Region 2 is stated as {circumflex over (n)}×{right arrow over (H)} ₂(ρ,φ,0)={right arrow over (J)} _(S),  (12) where {circumflex over (n)} is a unit normal in the positive vertical (+z) direction and {right arrow over (H)}₂ is the magnetic field strength in Region 2 expressed by Equation (1) above. Equation (12) implies that the fields specified in Equations (1)-(3) may be obtained by driving a radial surface current density along the boundary interface, such radial surface current density being specified by J _(ρ)(ρ′)=−AH ₁ ⁽²⁾(−jγρ′)  (13) where A is a constant yet to be determined. Further, it should be noted that close-in to the polyphase waveguide probe (for ρ<<λ), Equation (13) above has the behavior

$\begin{matrix} {{J_{close}\left( \rho^{\prime} \right)} = {\frac{- {A\left( {j\; 2} \right)}}{\pi\left( {{- j}\;\gamma\;\rho} \right)} = {{- H_{\phi}} = {- {\frac{I_{o}}{2\pi\;\rho^{\prime}}.}}}}} & (14) \end{matrix}$ One may wish to note the negative sign. This means that when source current flows vertically upward, the required “close-in” ground current flows radially inward. By field matching on H_(φ) “close-in” we find that

$\begin{matrix} {A = \frac{{- I_{o}}\gamma}{4}} & (15) \end{matrix}$ in Equations (1)-(6) and (13). Therefore, Equation (13) may be restated as

$\begin{matrix} {{J_{\rho}\left( \rho^{\prime} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\;{\gamma\rho}^{\prime}} \right)}.}}} & (16) \end{matrix}$

With reference then to FIG. 3, shown is an example of a polyphase waveguide probe 200 that includes a charge terminal T₁ and a charge terminal T₂ that are arranged along a vertical axis z. The polyphase waveguide probe 200 is disposed above a lossy conducting medium 203 according to an embodiment of the present disclosure. The lossy conducting medium 203 makes up Region 1 (FIG. 2) according to one embodiment. In addition, a second medium 206 shares a boundary interface with the lossy conducting medium 203 and makes up Region 2 (FIG. 2). The polyphase waveguide probe 200 includes a probe coupling circuit 209 that couples an excitation source 213 to the charge terminals T₁ and T₂ as is discussed in greater detail with reference to later figures.

The charge terminals T₁ and T₂ are positioned over the lossy conducting medium 203. The charge terminal T₁ may be considered a capacitor, and the charge terminal T₂ may comprise a counterpoise or lower capacitor as is described herein. According to one embodiment, the charge terminal T₁ is positioned at height H₁, and the charge terminal T₂ is positioned directly below T₁ along the vertical axis z at height H₂, where H₂ is less than H₁. The height h of the transmission structure presented by the polyphase waveguide probe 200 is h=H₁−H₂. Given the foregoing discussion, one can determine asymptotes of the radial Zenneck surface current on the surface of the lossy conducting medium J_(ρ)(ρ) to be J₁(ρ) close-in and J₂(ρ) far-out, where

Close-in (ρ<λ/8):

$\begin{matrix} {{{{J_{\rho}(\rho)} \sim J_{1}} = {\frac{I_{1} + I_{2}}{2\pi\;\rho} + \frac{{E_{\rho}^{QS}\left( Q_{1} \right)} + {E_{\rho}^{QS}\left( Q_{2} \right)}}{Z_{\rho}}}},} & (17) \end{matrix}$ and

Far-out (ρ>>λ/8):

$\begin{matrix} {{{J_{\rho}(\rho)} \sim J_{2}} = {\frac{j\;{\gamma\omega}\; Q_{1}}{4} \times \sqrt{\frac{2\gamma}{\pi}} \times \frac{{\mathbb{e}}^{{- {({\alpha + {j\;\beta}})}}\rho}}{\sqrt{\rho}}}} & (18) \end{matrix}$ where I₁ is the conduction current feeding the charge Q₁ on the first charge terminal T₁, and I₂ is the conduction current feeding the charge Q₂ on the second charge terminal T₂. The charge Q₁ on the upper charge terminal T₁ is determined by Q₁=C₁V₁, where C₁ is the isolated capacitance of the charge terminal T₁. Note that there is a third component to J₁ set forth above given by

$\frac{\left( E_{\rho}^{Q_{1}} \right)}{Z_{\rho}},$ which follows from the Leontovich boundary condition and is the radial current contribution in the lossy conducting medium 203 pumped by the quasi-static field of the elevated oscillating charge on the first charge terminal Q₁. The quantity

$Z_{\rho} = \frac{j\;\omega\;\mu_{o}}{\gamma_{e}}$ is the radial impedance of the lossy conducting medium, where γ_(e)=(jωμ₁σ₁−ω²μ₁∈₁)^(1/2).

The asymptotes representing the radial current close-in and far-out as set forth by Equations (17) and (18) are complex quantities. According to various embodiments, a physical surface current, J(r), is synthesized to match as close as possible the current asymptotes in magnitude and phase. That is to say close-in, |J(r)| is to be tangent to |J₁|, and far-out |J(r)| is to be tangent to |J₂|. Also, according to the various embodiments, the phase of J(r) should transition from the phase of J₁ close-in to the phase of J₂ far-out.

According to one embodiment, if any of the various embodiments of a polyphase waveguide probe described herein are adjusted properly, this configuration will give at least an approximate magnitude and phase match to the Zenneck mode and launch Zenneck surface waves. It should be noted that the phase far-out, φ₂, is proportional to the propagation phase corresponding to e^(−jβρ) plus a fixed “phase boost” due to the phase of √{square root over (γ)} which is arg(√{square root over (γ)}), jΦ ₂(ρ)=−jβρ+arg(√{square root over (γ)})  (19) where γ is expressed in Equation (9) above, and depending on the values for ∈_(r) and σ at the site of transmission on the lossy conducting medium and the operating frequency f, arg(√{square root over (γ)}), which has two complex roots, is typically on the order of approximately 45° or 225°. Stated another way, in order to match the Zenneck surface wave mode at the site of transmission to launch a Zenneck surface wave, the phase of the surface current |J₂| far-out should differ from the phase of the surface current |J₁| close-in by the propagation phase corresponding to e^(−jβ(ρ) ² ^(-ρ) ¹ ⁾ plus a constant of approximately 45 degrees or 225 degrees. This is because there are two roots for √{square root over (γ)}, one near π/4 and one near 5π/4. The properly adjusted synthetic radial surface current is

$\begin{matrix} {{J_{\rho}\left( {\rho,\phi,0} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\;{\gamma\rho}} \right)}.}}} & (20) \end{matrix}$ By Maxwell's equations, such a J(ρ) surface current automatically creates fields that conform to

$\begin{matrix} {{H_{\phi} = {\frac{{- \gamma}\; I_{o}}{4}{\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;{\gamma\rho}} \right)}}},} & (21) \\ {{E_{\rho} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{u_{2}}{j\;\omega\; ɛ_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},{and}} & (22) \\ {E_{z} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{- \gamma}{\omega\; ɛ_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}.}}} & (23) \end{matrix}$ Thus, the difference in phase between the surface current |J₂| far-out and the surface current |J₁| close-in for the Zenneck surface wave mode that is to be matched is due to the inherent characteristics of the Hankel functions in Equations (20)-(23) set forth above. It is of significance to recognize that the fields expressed by Equations (1)-(6) and (20) have the nature of a transmission line mode bound to a lossy interface, not radiation fields such as are associated with groundwave propagation. See Barlow, H. M. and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5. These fields automatically satisfy the complex Brewster angle requirement for zero reflection, which means that radiation is negligible, while surface guided wave propagation is dramatically enhanced, as is verified and supported in the experimental results provided below.

At this point, a review of the nature of the Hankel functions used in Equations (20)-(23) is provided with emphasis on a special property of these solutions of the wave equation. One might observe that the Hankel functions of the first and second kind and order n are defined as complex combinations of the standard Bessel functions of the first and second kinds H _(n) ⁽¹⁾(x)=J _(n)(x)+jN _(n)(x) and  (24) H _(n) ⁽²⁾(x)=J _(n)(x)−jN _(n)(x).  (25) These functions represent cylindrical waves propagating radially inward (superscript (1)) and outward (superscript (2)), respectively. The definition is analogous to the relationship e^(±jx)=cos x±j sin x. See, for example, Harrington, R. F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460-463.

That H_(n) ⁽²⁾(k_(ρ)ρ) is an outgoing wave is easily recognized from its large argument asymptotic behavior that is obtained directly from the series definitions of J_(n)(x) and N_(n)(x),

$\begin{matrix} {{{H_{n}^{(2)}(x)}\underset{x->\infty}{\longrightarrow}\sqrt{\frac{2j}{\pi\; x}}}j^{n}{\mathbb{e}}^{{- j}\; x}} & (26) \end{matrix}$ which, when multiplied by e^(jωt), is an outward propagating cylindrical wave of the form e^(j(ωt-kρ)) with a 1/√ρ spatial variation. The phase of the exponential component is ψ=(ωt−kρ). It is also evident that

$\begin{matrix} {{{{H_{n}^{(2)}(x)}\underset{x->\infty}{\longrightarrow}{\varphi j}^{n}}{H_{0}^{(2)}(x)}},} & (27) \end{matrix}$ and, a further useful property of Hankel functions is expressed by

$\begin{matrix} {{\frac{\partial{H_{0}^{(2)}(x)}}{\partial x} = {- {H_{1}^{(2)}(x)}}},} & (28) \end{matrix}$ which is described by Jahnke, E., and F. Emde, Tables of Functions, Dover, 1945, p. 145.

In addition, the small argument and large argument asymptotes of the outward propagating Hankel function are as follows:

$\begin{matrix} {{H_{1}^{(2)}(x)}\underset{x->0}{\longrightarrow}\frac{j\; 2}{\pi\; x}} & (29) \\ {{{{H_{1}^{(2)}(x)}\underset{x->\infty}{\longrightarrow}j}\sqrt{\frac{2j}{\pi\; x}}{\mathbb{e}}^{{- j}\; x}} = {\sqrt{\frac{2}{\pi\; x}}{{\mathbb{e}}^{- {j{({x - \frac{\pi}{2} - \frac{\pi}{4}})}}}.}}} & (30) \end{matrix}$

Note that these asymptotic expressions are complex quantities. Also, unlike ordinary sinusoidal functions, the behavior of complex Hankel functions differs in-close and far-out from the origin. When x is a real quantity, Equations (29) and (30) differ in phase by √{square root over (j)}, which corresponds to an extra phase advance or “phase boost” of 45° or, equivalently λ/8.

With reference to FIG. 4, to further illustrate the phase transition between J₁ (FIG. 3) and J₂ (FIG. 3) shown is an illustration of the phases of the surface currents J₁ close-in and J₂ far-out relative to a position of a polyphase waveguide probe 200 (FIG. 3). As shown in FIG. 4, there are three different observation points P₀, P₁, and P₂. A transition region is located between the observation point P₁ and observation point P₂. The observation point P₀ is located at the position of the polyphase waveguide probe 200. The observation point P₁ is positioned “close-in” at a distance R₁ from the observation point P₀ that places the observation point P₁ between the transition region 216 and the observation point P₀. The observation point P₂ is positioned “far-out” at a distance R₂ from the observation point P₀ beyond the transition region 216 as shown.

At observation point P₀, the magnitude and phase of the radial current J is expressed as |J_(P) ₀ |∠φ₀. At observation point P₁, the magnitude and phase of the radial current J is expressed as |J_(P) ₁ |∠φ₀−βR₁, where the phase shift of βR₁ is attributable to the distance R₁ between the observation points P₀ and P₁. At observation point P₂, the magnitude and phase of the radial current J is expressed as |J_(P) ₂ |∠φ₀−βR₁+φ_(Δ) where the phase shift of βR₁+φ_(Δ) is attributable to the distance R₂ between the observation points P₀ and P₂ AND an additional phase shift that occurs in the transition region 216. The additional phase shift φ_(Δ) occurs as a property of the Hankel function as mentioned above.

The foregoing reflects the fact that the polyphase waveguide probe 200 generates the surface current J₁ close-in and then transitions to the J₂ current far-out. In the transition region 216, the phase of the Zenneck surface-waveguide mode transitions by approximately 45 degrees or ⅛λ. This transition or phase shift may be considered a “phase boost” as the phase of the Zenneck surface-waveguide mode appears to advance by 45 degrees in the transition region 216. The transition region 216 appears to occur somewhere less than 1/10 of a wavelength of the operating frequency.

Referring back to FIG. 3, according to one embodiment, a polyphase waveguide probe may be created that will launch the appropriate radial surface current distribution. According to one embodiment, a Zenneck waveguide mode is created in a radial direction. If the J(r) given by Equation (20) can be created, it will automatically launch Zenneck surface waves.

In addition, further discussion is provided regarding the charge images Q₁′ and Q₂′ of the charges Q₁ and Q₂ on the charge terminals T₁ and T₂ of one example polyphase waveguide probe shown in FIG. 3. Analysis with respect to the lossy conducting medium assumes the presence of induced effective image charges Q₁′ and Q₂′ beneath the polyphase waveguide probes coinciding with the charges Q₁ and Q₂ on the charge reservoirs T₁ and T₂ as described herein. Such image charges Q₁′ and Q₂′ must also be considered in the analysis. These image charges Q₁′ and Q₂′ are not merely 180° out of phase with the primary source charges Q₁ and Q₂ on the charge reservoirs T₁ and T₂, as they would be in the case of a perfect conductor. A lossy conducting medium such as, for example, a terrestrial medium presents phase shifted images. That is to say, the image charges Q₁′ and Q₂′ are at complex depths. For a discussion of complex images, reference is made to Wait, J. R., “Complex Image Theory—Revisited,” IEEE Antennas and Propagation Magazine, Vol. 33, No. 4, August 1991, pp. 27-29, which is incorporated herein by reference in its entirety.

Instead of the image charges Q₁′ and Q₂′ being at a depth that is equal to the height of the charges Q₁ and Q₂ (i.e. z_(n)′=−h_(n)), a conducting mirror 215 is placed at depth z=−d/2 and the image itself appears at a “complex distance” (i.e., the “distance” has both magnitude and phase), given by z_(n)′=−D_(n)=−(d+h_(n))≠−h_(n), where n=1, 2, and for vertically polarized sources,

$\begin{matrix} {{d = {{\frac{2\sqrt{\gamma_{e}^{2} + k_{o}^{2}}}{\gamma_{e}^{2}} \approx \frac{2}{\gamma_{e}}} = {{d_{r} + {j\; d_{i}}} = {{d}{\angle\zeta}}}}},{where}} & (31) \\ {{\gamma_{e}^{2} = {{j\;\omega\;\mu_{1}\sigma_{1}} - {\omega^{2}\mu_{1}ɛ_{1}}}},{and}} & (32) \\ {k_{o} = {\omega{\sqrt{\mu_{o}ɛ_{o}}.}}} & (33) \end{matrix}$

The complex spacing of image charges Q₁′ and Q₂′, in turn, implies that the external fields will experience extra phase shifts not encountered when the interface is either a lossless dielectric or a perfect conductor. The essence of the lossy dielectric image-theory technique is to replace the finitely conducting Earth (or lossy dielectric) by a perfect conductor located at a complex depth, z=−d/2. Next, a source image is then located at a complex depth D_(n)=d/2+d/2+h_(n)=d+h_(n), where n=1, 2. Thereafter, one can calculate the fields above ground (z≧0) using a superposition of the physical charge (at z=+h) plus its image (at z′=−D). The charge images Q₁′ and Q₂′ at complex depths actually assist in obtaining the desired current phases specified in Equations (20) and (21) above.

From Equations (2) and (3) above, it is noted that the ratio of E_(2z) to E_(2ρ) in Region 2 is given by

$\begin{matrix} {\frac{E_{2z}}{E_{2\rho}} = {\frac{{A\left( \frac{- \gamma}{\omega\; ɛ_{o}} \right)}{\mathbb{e}}^{{- u_{2}}z}{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}{{A\left( \frac{u_{2}}{j\;\omega\; ɛ_{o}} \right)}{\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}} = {\left( \frac{{- j}\;\gamma}{u_{2}} \right){\frac{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}.}}}} & (34) \end{matrix}$ Also, it should be noted that asymptotically,

$\begin{matrix} {{{H_{n}^{(2)}(x)}\underset{x->\infty}{\longrightarrow}j^{n}}{{H_{0}^{(2)}(x)}.}} & (35) \end{matrix}$ Consequently, it follows directly from Equations (2) and (3) that

$\begin{matrix} {{\frac{E_{2z}}{E_{2\rho}} = {\sqrt{ɛ_{r} - {j\; x}} = {n = {\tan\;\psi_{i,B}}}}},} & (36) \end{matrix}$ where ψ_(i,B) is the complex Brewster angle. By adjusting source distributions and synthesizing complex Brewster angle illumination at the surface of a lossy conducting medium 203, Zenneck surface waves may be excited.

With reference to FIG. 5, shown is an incident field E polarized parallel to a plane of incidence. The electric field vector E is to be synthesized as an incoming non-uniform plane wave, polarized parallel to the plane of incidence. The electric field vector E may be created from independent horizontal and vertical components as: {right arrow over (E)}(θ_(o))=E _(ρ) {circumflex over (ρ)}+E _(z) {circumflex over (z)}.  (37) Geometrically, the illustration in FIG. 5 suggests: E _(ρ)(ρ,z)=E(ρ,z)cos ψ_(o), and  (38a)

$\begin{matrix} {{{E_{z}\left( {\rho,z} \right)} = {{{E\left( {\rho,z} \right)}{\cos\left( {\frac{\pi}{2} - \psi_{o}} \right)}} = {{E\left( {\rho,z} \right)}\sin\;\psi_{o}}}},} & \left( {38b} \right) \end{matrix}$ which means that the field ratio is

$\begin{matrix} {\frac{E_{z}}{E_{\rho}} = {\tan\;{\psi_{o}.}}} & (39) \end{matrix}$ However, recall that from Equation (36), tan θ_(i,B)=√{square root over (∈_(r) −jx)}  (40) so that, for a Zenneck surface wave, we desire ψ_(o)=θ_(i,B), which results in

$\begin{matrix} {\frac{E_{z}}{E_{\rho}} = {{\tan\;\psi_{o}} = {\sqrt{ɛ_{r} - {j\frac{\sigma}{\omega\; ɛ_{o}}}}.}}} & (41) \end{matrix}$

The Equations mean that if one controls the magnitude of the complex field ratio and the relative phase between the incident vertical and horizontal components E_(z) and E_(ρ) in a plane parallel to the plane of incidence, then the synthesized E-field vector will effectively be made to be incident at a complex Brewster angle. Such a circumstance will synthetically excite a Zenneck surface wave over the interface between Region 1 and Region 2.

With reference to FIG. 6, shown is another view of the polyphase waveguide probe 200 disposed above a lossy conducting medium 203 according to an embodiment of the present disclosure. The lossy conducting medium 203 makes up Region 1 (FIG. 2) according to one embodiment. In addition, a second medium 206 shares a boundary interface with the lossy conducting medium 203 and makes up Region 2 (FIG. 2).

According to one embodiment, the lossy conducting medium 203 comprises a terrestrial medium such as the planet Earth. To this end, such a terrestrial medium comprises all structures or formations included thereon whether natural or man-made. For example, such a terrestrial medium may comprise natural elements such as rock, soil, sand, fresh water, sea water, trees, vegetation, and all other natural elements that make up our planet. In addition, such a terrestrial medium may comprise man-made elements such as concrete, asphalt, building materials, and other man-made materials. In other embodiments, the lossy conducting medium 203 may comprise some medium other than the Earth, whether naturally occurring or man-made. In other embodiments, the lossy conducting medium 203 may comprise other media such as man-made surfaces and structures such as automobiles, aircraft, man-made materials (such as plywood, plastic sheeting, or other materials) or other media.

In the case that the lossy conducting medium 203 comprises a terrestrial medium or Earth, the second medium 206 may comprise the atmosphere above the ground. As such, the atmosphere may be termed an “atmospheric medium” that comprises air and other elements that make up the atmosphere of the Earth. In addition, it is possible that the second medium 206 may comprise other media relative to the lossy conducting medium 203.

The polyphase waveguide probe 200 comprises a pair of charge terminals T₁ and T₂. Although two charge terminals T₁ and T₂ are shown, it is understood that there may be more than two charge terminals T₁ and T₂. According to one embodiment, the charge terminals T₁ and T₂ are positioned above the lossy conducting medium 203 along a vertical axis z that is normal to a plane presented by the lossy conducting medium 203. In this respect, the charge terminal T₁ is placed directly above the charge terminal T₂ although it is possible that some other arrangement of two or more charge terminals T_(N) may be used. According to various embodiments, charges Q₁ and Q₂ may be imposed on the respective charge terminals T₁ and T₂.

The charge terminals T₁ and/or T₂ may comprise any conductive mass that can hold an electrical charge. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. The charge terminals T₁ and/or T₂ may comprise any shape such as a sphere, a disk, a cylinder, a cone, a torus, a randomized shape, or any other shape. Also note that the charge terminals T₁ and T₂ need not be identical, but each can have a separate size and shape, and be comprises of different conducting materials. According to one embodiment, the shape of the charge terminal T₁ is specified to hold as much charge as practically possible. Ultimately, the field strength of a Zenneck surface wave launched by a polyphase waveguide probe 200 is directly proportional to the quantity of charge on the terminal T₁.

If the charge terminals T₁ and/or T₂ are spheres or disks, the respective self-capacitance C₁ and C₂ can be calculated. For example, the self-capacitance of an isolated conductive sphere is C=4π∈_(o)r, where r comprises the radius of the sphere in meters. The self-capacitance of an isolated disk is C=8∈_(o)r, where r comprises the radius of the disk in meters.

Thus, the charge Q₁ stored on the charge terminal T₁ may be calculated as Q₁=C₁V, given the self-capacitance C₁ of the charge reservoir T₁ and voltage V that is applied to the charge terminal T₁.

With further reference to FIG. 6, according to one embodiment, the polyphase waveguide probe 200 comprises a probe coupling circuit 209 that is coupled to the charge terminals T₁ and T₂. The probe coupling circuit 209 facilitates coupling the excitation source 213 to the charge terminals T₁ and T₂, and facilitates generating respective voltage magnitudes and phases on the charge terminals T₁ and T₂ for a given frequency of operation. If more than two charge terminals T_(N) are employed, then the probe coupling circuit 209 would be configured to facilitate the generation of various voltage magnitudes and phases on the respective charge terminals T_(N) relative to each other. In the embodiment of the polyphase waveguide probe 200, the probe coupling circuit 209 comprises various circuit configurations as will be described.

In one embodiment, the probe coupling circuit 209 is specified so as to make the polyphase waveguide probe 200 electrically half-wave resonant. This imposes a voltage +V on a first one of the terminals T₁ or T₂, and a −V on the second one of the charge terminals T₁ or T₂ at any given time. In such case, the voltages on the respective charge terminals T₁ and T₂ are 180 degrees out of phase as can be appreciated. In the case that the voltages on the respective charge terminals T₁ and T₂ are 180 degrees out of phase, the largest voltage magnitude differential is experienced on the charge terminals T₁ and T₂. Alternatively, the probe coupling circuit 209 may be configured so that the phase differential between the charge terminals T₁ and T₂ is other than 180 degrees. To this end, the probe coupling circuit 209 may be adjusted to alter the voltage magnitudes and phases during adjustment of the polyphase waveguide probe 200.

By virtue of the placement of the charge terminal T₁ directly above the charge terminal T₂, a mutual capacitance C_(M) is created between the charge terminals T₁ and T₂. Also, the charge terminal T₁ has self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂ as mentioned above. There may also be a bound capacitance between the charge terminal T₁ and the lossy conducting medium 203, and a bound capacitance between the charge terminal T₂ and the lossy conducting medium 203, depending on the respective heights of the charge terminals T₁ and T₂. The mutual capacitance C_(M) depends on the distance between the charge terminals T₁ and T₂.

Ultimately, the field strength generated by the polyphase waveguide probe 200 will be directly proportional to the magnitude of the charge Q₁ that is imposed on the upper terminal T₁. The charge Q₁ is, in turn, proportional to the self-capacitance C₁ associated with the charge terminal T₁ since Q₁=C₁V, where V is the voltage imposed on the charge terminal T₁.

According to one embodiment, an excitation source 213 is coupled to the probe coupling circuit 209 in order to apply a signal to the polyphase waveguide probe 200. The excitation source 213 may be any suitable electrical source such as a voltage or current source capable of generating the voltage or current at the operating frequency that is applied to the polyphase waveguide probe 200. To this end, the excitation source 213 may comprise, for example, a generator, a function generator, transmitter, or other electrical source.

In one embodiment, the excitation source 213 may be coupled to the polyphase waveguide probe 200 by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling as will be described. In some embodiments, the probe coupling circuit 209 may be coupled to the lossy conducting medium 203. Also, in various embodiments, the excitation source 213 maybe coupled to the lossy conducting medium 203 as will be described.

In addition, it should be noted that, according to one embodiment, the polyphase waveguide probe 200 described herein has the property that its radiation resistance R_(r) is very small or even negligible. One should recall that radiation resistance R_(r) is the equivalent resistance that would dissipate the same amount of power that is ultimately radiated from an antenna. According to the various embodiments, the polyphase waveguide probe 200 launches a Zenneck surface wave that is a guided electromagnetic wave. According to the various embodiments, the polyphase waveguide probes described herein have little radiation resistance R_(r) because the height of such polyphase waveguide probes is usually small relative to their operating wavelengths. Stated another way, according to one embodiment, the polyphase waveguide probes described herein are “electrically small.” As contemplated herein, the phrase “electrically small” is defined as a structure such as the various embodiments of polyphase waveguide probes described herein that can be physically bounded by a sphere having a radius equal to λ/2π, where λ is the free-space wavelength. See Fujimoto, K., A. Henderson, K. Hirasawa, and J. R. James, Small Antennas, Wiley, 1987, p. 4.

To discuss further, the radiation resistance R_(r) for a short monopole antenna is expressed by

$\begin{matrix} {R_{r} = {160\mspace{14mu}{\pi^{2}\left( \frac{h}{\lambda} \right)}^{2}}} & (42) \end{matrix}$ where the short monopole antenna has a height h with a uniform current distribution, and where λ is the wavelength at the frequency of operation. See Stutzman, W. L. et al., “Antenna Theory and Design,” Wiley & Sons, 1981, p. 93.

Given that the value of the radiation resistance R_(r) is determined as a function of

$\left( \frac{h}{\lambda} \right)^{2},$ it follows that if the height h of the structure is small relative to the wavelength of the operating signal at the operating frequency, then the radiation resistance R_(r) will also be small. As one example, if the height h of the transmission structure is 10% of the wavelength of the operating signal at the operating frequency, then the resulting value of

$\left( \frac{h}{\lambda} \right)^{2}$ would be (0.1)²=0.01. It would follow that the radiation resistance R_(r) is correspondingly small.

Thus, according to various embodiments, if the effective height h of the transmission structure is less than or equal to

$\frac{\lambda}{2\pi},$ where λ is the wavelength at the operating frequency, then the radiation resistance R_(r) will be relatively small. For the various embodiments of polyphase waveguide probes 200 described below, the height h of the transmission structure may be calculated as h=H₁−H₂, where H₁ is the height of the charge terminal T₁, and H₂ is the height of the charge terminal T₂. It should be appreciated that the height h of the transmission structure for each embodiment of the polyphase waveguide probes 200 described herein can be determined in a similar manner.

While

$\frac{\lambda}{2\pi}$ is provided as one benchmark, it is understood that the ratio of the height h of the transmission structure over the wavelength of the operating signal at the operating frequency can be any value. However, it is understood that, at a given frequency of operation, as the height of a given transmission structure increases, the radiation resistance R_(r) will increase accordingly.

Depending on the actual values for the height h and the wavelength of the operating signal at the operating frequency, it is possible that the radiation resistance R_(r) may be of a value such that some amount of radiation may occur along with the launching of a Zenneck surface wave. To this end, the polyphase waveguide probe 200 can be constructed to have a short height h relative to the wavelength at the frequency of operation so as to ensure that little or substantially zero energy is lost in the form of radiation.

In addition, the placement of the charge reservoirs T₁ and T₂ along the vertical axis z provides for symmetry in the Zenneck surface wave that is launched by the polyphase waveguide probe 200 as described by the Hankel functions in Equations (20)-(23) set forth above. Although the polyphase waveguide probe 200 is shown with two charge reservoirs T₁ and T₂ along the vertical axis z that is normal to a plane making up the surface of the lossy conducting medium 203, it is understood that other configurations may be employed that will also provide for the desired symmetry. For example, additional charge reservoirs T_(N) may be positioned along the vertical axis z, or some other arrangement may be employed. In some embodiments, symmetry of transmission may not be desired. In such case, the charge reservoirs T_(N) may be arranged in a configuration other than along a vertical axis z to provide for an alternative transmission distribution pattern.

When properly adjusted to operate at a predefined operating frequency, the polyphase waveguide probe 200 generates a Zenneck surface wave along the surface of the lossy conducting medium 203. To this end, an excitation source 213 may be employed to generate electrical energy at a predefined frequency that is applied to the polyphase waveguide probe 200 to excite the structure. The energy from the excitation source 213 is transmitted in the form of a Zenneck surface wave by the polyphase waveguide probe 200 to one or more receivers that are also coupled to the lossy conducting medium 203 or that are located within an effective transmission range of the polyphase waveguide probe 200. The energy is thus conveyed in the form of a Zenneck surface wave which is a surface-waveguide mode or a guided electromagnetic field. In the context of modern power grids using high voltage lines, a Zenneck surface wave comprises a transmission line mode.

Thus, the Zenneck surface wave generated by the polyphase waveguide probe 200 is not a radiated wave, but a guided wave in the sense that these terms are described above. The Zenneck surface wave is launched by virtue of the fact that the polyphase waveguide probe 200 creates electromagnetic fields that are substantially mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium 203. When the electromagnetic fields generated by the polyphase waveguide probe 200 are substantially mode-matched as such, the electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium 203 that results in little or no reflection. Note that if the polyphase waveguide probe 200 is not substantially mode-matched to the Zenneck surface wave mode, then a Zenneck surface wave will not be launched since the complex Brewster angle of the lossy conducting medium 203 would not have been attained.

In the case that the lossy conducting medium 203 comprises a terrestrial medium such as the Earth, the Zenneck surface wave mode will depend upon the dielectric permittivity ∈_(r) and conductivity a of the site at which the polyphase waveguide probe 200 is located as indicated above in Equations (1)-(11). Thus, the phase of the Hankel functions in Equations (20)-(23) above depends on these constitutive parameters at the launch site and on the frequency of operation.

In order to excite the fields associated with the Zenneck surface wave mode, according to one embodiment, the polyphase waveguide probe 200 substantially synthesizes the radial surface current density on the lossy conducting medium of the Zenneck surface wave mode as is expressed by Equation (20) set forth above. When this occurs, the electromagnetic fields are then substantially or approximately mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium 203. To this end, the match should be as close as is practicable. According to one embodiment, this Zenneck surface wave mode to which the electromagnetic fields are substantially matched is expressed in Equations (21)-(23) set forth above.

In order to synthesize the radial surface current density in the lossy conducting medium of the Zenneck surface wave mode, the electrical characteristics of the polyphase waveguide probe 200 should be adjusted to impose appropriate voltage magnitudes and phases on the charge terminals T₁ and T₂ for a given frequency of operation and given the electrical properties of the site of transmission. If more than two charge terminals T_(N) are employed, then appropriate voltage magnitudes and phases would need to be imposed on the respective charge terminals T_(N), where N may even be a very large number effectively comprising a continuum of charge terminals.

In order to obtain the appropriate voltage magnitudes and phases for a given design of a polyphase waveguide probe 200 at a given location, an iterative approach may be used. Specifically, analysis may be performed of a given excitation and configuration of a polyphase waveguide probe 200 taking into account the feed currents to the terminals T₁ and T₂, the charges on the charge terminals T₁ and T₂, and their images in the lossy conducting medium 203 in order to determine the radial surface current density generated. This process may be performed iteratively until an optimal configuration and excitation for a given polyphase waveguide probe 200 is determined based on desired parameters. To aid in determining whether a given polyphase waveguide probe 200 is operating at an optimal level, a guided field strength curve 103 (FIG. 1) may be generated using Equations (1)-(11) above based on values for the conductivity of Region 1 (σ₁) and the permittivity of Region 1 (∈₁) at the location of the polyphase waveguide probe 200. Such a guided field strength curve 103 will provide a benchmark for operation such that measured field strengths can be compared with the magnitudes indicated by the guided field strength curve 103 to determine if optimal transmission has been achieved.

In order to arrive at an optimized polyphase waveguide probe 200, various parameters associated with the polyphase waveguide probe 200 may be adjusted. Stated another way, the various parameters associated with the polyphase waveguide probe 200 may be varied to adjust the polyphase waveguide probe 200 to a desired operating configuration.

One parameter that may be varied to adjust the polyphase waveguide probe 200 is the height of one or both of the charge terminals T₁ and/or T₂ relative to the surface of the lossy conducting medium 203. In addition, the distance or spacing between the charge terminals T₁ and T₂ may also be adjusted. In doing so, one may minimize or otherwise alter the mutual capacitance C_(M) or any bound capacitances between the charge terminals T₁ and T₂ and the lossy conducting medium 203 as can be appreciated.

Alternatively, another parameter that can be adjusted is the size of the respective charge terminals T₁ and/or T₂. By changing the size of the charge terminals T₁ and/or T₂, one will alter the respective self-capacitances C₁ and/or C₂, and the mutual capacitance C_(M) as can be appreciated. Also, any bound capacitances that exist between the charge terminals T₁ and T₂ and the lossy conducting medium 203 will be altered. In doing so, the voltage magnitudes and phases on the charge terminals T₁ and T₂ are altered.

Still further, another parameter that can be adjusted is the probe coupling circuit 209 associated with the polyphase waveguide probe 200. This may be accomplished by adjusting the size of the inductive and/or capacitive reactances that make up the probe coupling circuit 209. For example, where such inductive reactances comprise coils, the number of turns on such coils may be adjusted. Ultimately, the adjustments to the probe coupling circuit 209 can be made to alter the electrical length of the probe coupling circuit 209, thereby affecting the voltage magnitudes and phases on the charge terminals T₁ and T₂.

It is also the case that one can adjust the frequency of an excitation source 213 applied to the polyphase waveguide probe 200 to optimize the transmission of a Zenneck surface wave. However, if one wishes to transmit at a given frequency, other parameters would need to be adjusted to optimize transmission.

Note that the iterations of transmission performed by making the various adjustments may be implemented by using computer models or by adjusting physical structures as can be appreciated. In one approach, a field meter tuned to the transmission frequency may be placed an appropriate distance from the polyphase waveguide probe 200 and adjustments may be made as set forth above until a maximum or any other desired field strength of a resulting Zenneck surface wave is detected. To this end, the field strength may be compared with a guided field strength curve 103 (FIG. 1) generated at the desired operating frequency and voltages on the terminals T₁ and T₂. According to one approach, the appropriate distance for placement of such a field meter may be specified as greater than the transition region 216 (FIG. 4) in the “far-out” region described above where the surface current J₂ dominates.

By making the above adjustments, one can create corresponding “close-in” surface current J₁ and “far-out” surface current J₂ that approximate the same currents J(r) of the Zenneck surface wave mode specified in Equations (17) and (18) set forth above. In doing so, the resulting electromagnetic fields would be substantially or approximately mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium 203.

Referring next to FIGS. 7A through 7J, shown are additional examples of polyphase waveguide probes 200, denoted herein as polyphase waveguide probes 200 a-j, according to various embodiments of the present disclosure. The polyphase waveguide probes 200 a-j each include a different probe coupling circuit 209, denoted herein as probe coupling circuits 209 a-j, according to various embodiments. While several examples of probe coupling circuits 209 a-j are described, it is understood that these embodiments are merely examples and that there may be many other probe coupling circuits 209 not set forth herein that may be employed to provide for the desired voltage magnitudes and phases on the charge terminals T₁ and T₂ according to the principles set forth herein in order to facilitate the launching of Zenneck surface waves.

In addition, each of the probe coupling circuits 209 a-j may employ, but are not limited to, inductive impedances comprising coils. Even though coils are used, it is understood that other circuit elements, both lumped and distributed, may be employed as reactances. Also, other circuit elements may be included in the probe coupling circuits 209 a-j beyond those illustrated herein. In addition, it is noted that the various polyphase waveguide probes 200 a-j with their respective probe coupling circuits 209 a-j are merely described herein to provide examples. To this end, there may be many other polyphase waveguide probes 200 that employ various probe coupling circuits 209 and other circuitry that can be used to launch Zenneck surface waves according to the various principles set forth herein.

Referring now to FIG. 7A, shown is a first example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 a, according to one embodiment. The polyphase waveguide probe 200 a includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 a includes a probe coupling circuit 209 a that comprises an inductive impedance comprising a coil L_(1a) having a pair of leads that are coupled to respective ones of the charge terminals T₁ and T₂. In one embodiment, the coil L_(1a) is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the polyphase waveguide probe 200 a.

While the electrical length of the coil L_(1a) is specified as approximately one-half (½) the wavelength at the operating frequency, it is understood that the coil L_(1a) may be specified with an electrical length at other values. According to one embodiment, the fact that the coil L_(1a) has an electrical length of approximately one-half the wavelength at the operating frequency provides for an advantage in that a maximum voltage differential is created on the charge terminals T₁ and T₂. Nonetheless, the length or diameter of the coil L_(1a) may be increased or decreased when adjusting the polyphase waveguide probe 200 a to obtain optimal excitation of a Zenneck surface wave mode. Alternatively, it may be the case that the inductive impedance is specified to have an electrical length that is significantly less than or greater than ½ the wavelength at the operating frequency of the polyphase waveguide probe 200 a.

According to one embodiment, the excitation source 213 is coupled to the probe coupling circuit 209 by way of magnetic coupling. Specifically, the excitation source 213 is coupled to a coil L_(P) that is inductively coupled to the coil L_(1a). This may be done by link coupling, a tapped coil, a variable reactance, or other coupling approach as can be appreciated. To this end, the coil L_(P) acts as a primary, and the coil L_(1a) acts as a secondary as can be appreciated.

In order to adjust the polyphase waveguide probe 200 a for the transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of the coil L_(1a) may be altered by adding or eliminating turns or by changing some other dimension of the coil L_(1a).

Based on experimentation with the polyphase waveguide probe 200 a, this appears to be the easiest of the polyphase waveguide probes 200 a-j to adjust and to operate to achieve a desired efficiency.

Referring now to FIG. 7B, shown is an example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 b, according to one embodiment. The polyphase waveguide probe 200 b includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminals T₁ and T₂ are positioned along a vertical axis z to provide for cylindrical symmetry in the resulting Zenneck surface wave as was described above. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 b also includes a probe coupling circuit 209 b comprising a first coil L_(1b) and a second coil L_(2b). The first coil L_(1b) is coupled to each of the charge terminals T₁ and T₂ as shown. The second coil L_(2b) is coupled to the charge terminal T₂ and to the lossy conducting medium 203.

The excitation source 213 is magnetically coupled to the probe coupling circuit 209 b in a manner similar as was mentioned with respect to the polyphase waveguide probe 200 a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil L_(P) that acts as a primary and the coil L_(1b) acts as a secondary. Alternatively, the coil L_(2b) may act as a secondary as well.

In order to adjust the polyphase waveguide probe 200 b for the transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of each of the coils L_(1b) and L_(2b) may be altered by adding or eliminating turns or by changing some other dimension of the respective coils L_(1b) or L_(2b).

Referring now to FIG. 7C, shown is another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 c, according to one embodiment. The polyphase waveguide probe 200 c includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 c also includes a probe coupling circuit 209 c comprising a coil L_(1c). One end of the coil L_(1c) is coupled to the charge terminal T₁ as shown. The second end of the coil L_(1c) is coupled to the lossy conducting medium 203. A tap that is coupled to the charge terminal T₂ is positioned along the coil L_(1c).

The excitation source 213 is magnetically coupled to the probe coupling circuit 209 c in a manner similar as was mentioned with respect to the polyphase waveguide probe 200 a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil L_(P) that acts as a primary and the coil L_(1c) acts as a secondary. The coil L_(P) can be positioned at any location along the coil L_(1c).

In order to adjust the polyphase waveguide probe 200 c for the excitation and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of the coil L_(1c) may be altered by adding or eliminating turns, or by changing some other dimension of the coil L_(1c). In addition, the inductance presented by the portions of the coil L_(1c) above and below the tap may be adjusted by moving the position of the tap.

Referring now to FIG. 7D, shown is yet another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 d, according to one embodiment. The polyphase waveguide probe 200 d includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 d also includes a probe coupling circuit 209 d comprising a first coil L_(1d) and a second coil L_(2d). A first lead of the first coil L_(1d) is coupled to the charge terminal T₁, and the second lead of the first coil L_(1d) is coupled to the lossy conducting medium 203. A first lead of the second coil L_(2d) is coupled to the charge terminal T₂, and the second lead of the second coil L_(2d) is coupled to the lossy conducting medium 203.

The excitation source 213 is magnetically coupled to the probe coupling circuit 209 d in a manner similar as was mentioned with respect to the polyphase waveguide probe 200 a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil L_(P) that acts as a primary and the coil L_(2d) acts as a secondary. Alternatively, the coil L_(1d) may act as a secondary as well.

In order to adjust the polyphase waveguide probe 200 d for the excitation and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of each of the coils L_(1d) and L_(2d) may be altered by adding or eliminating turns or by changing some other dimension of the respective coils L_(1d) or L_(2d).

Referring now to FIG. 7E, shown is still another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 e, according to one embodiment. The polyphase waveguide probe 200 e includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminals T₁ and T₂ are positioned along a vertical axis z to provide for cylindrical symmetry in the resulting Zenneck surface wave as was described above. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 e also includes a probe coupling circuit 209 e comprising a first coil L_(1e) and a resistor R₂. A first lead of the first coil L_(1e) is coupled to the charge terminal T₁, and the second lead of the first coil L_(1e) is coupled to the lossy conducting medium 203. A first lead of the resistor R₂ is coupled to the charge terminal T₂, and the second lead of the resistor R₂ is coupled to the lossy conducting medium 203.

The excitation source 213 is magnetically coupled to the probe coupling circuit 209 e in a manner similar as was mentioned with respect to the polyphase waveguide probe 200 a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil L_(P) that acts as a primary and the coil L_(1e) acts as a secondary.

In order to adjust the polyphase waveguide probe 200 e for the transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of the coil L_(1e) may be altered by adding or eliminating turns or by changing some other dimension of the respective coils L_(1e). In addition, the magnitude of the resistance R₂ may be adjusted as well.

Referring now to FIG. 7F, shown is a further example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 f, according to one embodiment. The polyphase waveguide probe 200 f includes a charge terminal T₁ and a ground screen G that acts as a second charge terminal. The charge terminal T₁ and the ground screen G are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. Note that to calculate the height h of the transmission structure, the height H₂ of the ground screen G is subtracted from the height H₁ of the charge terminal T₁.

The charge terminal T₁ has a self-capacitance C₁, and the ground screen G has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminal T₁ and the ground screen G, respectively, depending on the voltages applied to the charge terminal T₁ and the ground screen G at any given instant. A mutual capacitance C_(M) may exist between the charge terminal T₁ and the ground screen G depending on the distance there between. In addition, bound capacitances may exist between the charge terminal T₁ and/or the ground screen G and the lossy conducting medium 203 depending on the heights of the charge terminal T₁ and the ground screen G with respect to the lossy conducting medium 203. Generally, a bound capacitance will exist between the ground screen G and the lossy conducting medium 203 due to its proximity to the lossy conducting medium 203.

The polyphase waveguide probe 200 f includes a probe coupling circuit 209 f that is made up of an inductive impedance comprising a coil L_(1f) having a pair of leads that are coupled to the charge terminal T₁ and ground screen G. In one embodiment, the coil L_(1f) is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the polyphase waveguide probe 200 f.

While the electrical length of the coil L_(1f) is specified as approximately one-half (½) the wavelength at the operating frequency, it is understood that the coil L_(1f) may be specified with an electrical length at other values. According to one embodiment, the fact that the coil L_(1f) has an electrical length of approximately one-half the wavelength at the operating frequency provides for an advantage in that a maximum voltage differential is created on the charge terminal T₁ and the ground screen G. Nonetheless, the length or diameter of the coil L_(1f) may be increased or decreased when adjusting the polyphase waveguide probe 200 f to obtain optimal transmission of a Zenneck surface wave. Alternatively, it may be the case that the inductive impedance is specified to have an electrical length that is significantly less than or greater than ½ the wavelength at the operating frequency of the polyphase waveguide probe 200 f.

According to one embodiment, the excitation source 213 is coupled to the probe coupling circuit 209 f by way of magnetic coupling. Specifically, the excitation source 213 is coupled to a coil L_(P) that is inductively coupled to the coil L_(1f). This may be done by link coupling, a phasor/coupling network, or other approach as can be appreciated. To this end, the coil L_(P) acts as a primary, and the coil L_(1f) acts as a secondary as can be appreciated.

In order to adjust the polyphase waveguide probe 200 a for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of the coil L_(1f) may be altered by adding or eliminating turns or by changing some other dimension of the coil L_(1f).

Referring now to FIG. 7G, shown is another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 g, according to one embodiment. The polyphase waveguide probe 200 g includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminals T₁ and T₂ are positioned along a vertical axis z to provide for cylindrical symmetry in the resulting Zenneck surface wave as was described above. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 g also includes a probe coupling circuit 209 g comprising a first coil L_(1g), a second coil L_(2g), and a variable capacitor C_(V) The first coil L_(1g) is coupled to each of the charge terminals T₁ and T₂ as shown. The second coil L_(2g) has a first lead that is coupled to a variable capacitor C_(V) and a second lead that is coupled to the lossy conducting medium 203. The variable capacitor C_(V), in turn, is coupled to the charge terminal T₂ and the first coil L_(1g).

The excitation source 213 is magnetically coupled to the probe coupling circuit 209 g in a manner similar as was mentioned with respect to the polyphase waveguide probe 200 a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil L_(P) that acts as a primary and either the coil L_(1g) or the coil L_(2g) may act as a secondary.

In order to adjust the polyphase waveguide probe 200 g for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of each of the coils L_(1g) and L_(2g) may be altered by adding or eliminating turns or by changing some other dimension of the respective coils L_(1g) or L_(2g). In addition, the variable capacitance C_(V) may be adjusted.

Referring now to FIG. 7H, shown is yet another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 h, according to one embodiment. The polyphase waveguide probe 200 h includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 h also includes a probe coupling circuit 209 h comprising a first coil L_(1h) and a second coil L_(2h). The first lead of the first coil L_(1h) is coupled to the charge terminal T₁, and the second lead of the first coil L_(1h) is coupled to the second charge terminal T₂. A first lead of the second coil L_(2h) is coupled to a terminal T_(T), and the second lead of the second coil L_(2h) is coupled to the lossy conducting medium 203. The terminal T_(T) is positioned relative to the charge terminal T₂ such that a coupling capacitance C_(C) exists between the charge terminal T₂ and the terminal T_(T).

The excitation source 213 is magnetically coupled to the probe coupling circuit 209 h in a manner similar as was mentioned with respect to the polyphase waveguide probe 200 a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil L_(P) that acts as a primary and the coil L_(2h) acts as a secondary. Alternatively, the coil L_(1h) may act as a secondary as well.

In order to adjust the polyphase waveguide probe 200 h for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of each of the coils L_(1h) and L_(2h) may be altered by adding or eliminating turns or by changing some other dimension of the respective coils L_(1h) or L_(2h). Also the spacing between the charge terminal T₂ and the terminal T_(T) may be altered, thereby modifying the coupling capacitance C_(C) as can be appreciated.

Referring now to FIG. 7I, shown is yet another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 i, according to one embodiment. The polyphase waveguide probe 200 i is very similar to the polyphase waveguide probe 200 h (FIG. 7H) except for the fact that the excitation source 213 is series-coupled to the probe coupling circuit 209 i as will be described.

To this end, the polyphase waveguide probe 200 i includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 i also includes a probe coupling circuit 209 i comprising a first coil L_(1i) and a second coil L_(2i). The first lead of the first coil L_(1i) is coupled to the charge terminal T₁, and the second lead of the first coil L_(1i) is coupled to the second charge terminal T₂. A first lead of the second coil L_(2i) is coupled to a terminal T_(T), and the second lead of the second coil L_(2i) is coupled to an output of the excitation source 213. Also, a ground lead of the excitation source 213 is coupled to the lossy conducting medium 203. The terminal T_(T) is positioned relative to the charge terminal T₂ such that a coupling capacitance C_(C) exists between the charge terminal T₂ and the terminal T_(T).

The polyphase waveguide probe 200 i provides one example where the excitation source 213 is series-coupled to the probe coupling circuit 209 i as mentioned above. Specifically, the excitation source 213 is coupled between the coil L_(2i) and the lossy conducting medium 203.

In order to adjust the polyphase waveguide probe 200 i for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of each of the coils L_(1i) and L_(2i) may be altered by adding or eliminating turns or by changing some other dimension of the respective coils L_(1i) or L_(2i). Also the spacing between the charge terminal T₂ and the terminal T_(T) may be altered, thereby modifying the coupling capacitance C_(C) as can be appreciated.

Referring now to FIG. 7J, shown is an example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200 j, according to one embodiment. The polyphase waveguide probe 200 j includes the charge terminals T₁ and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. In this embodiment, the charge terminal T₁ comprises a sphere and the charge terminal T₂ comprises a disk. In this respect, the polyphase waveguide probe 200 j provides an illustration that the charge terminals T_(N) may comprise any shape.

The charge terminal T₁ has a self-capacitance C₁, and the charge terminal T₂ has a self-capacitance C₂. During operation, charges Q₁ and Q₂ are imposed on the charge terminals T₁ and T₂, respectively, depending on the voltages applied to the charge terminals T₁ and T₂ at any given instant. A mutual capacitance C_(M) may exist between the charge terminals T₁ and T₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T₁ and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals T₁ and T₂ with respect to the lossy conducting medium 203.

The polyphase waveguide probe 200 j includes a probe coupling circuit 209 j comprising an inductive impedance comprising a coil L_(1j) having a pair of leads that are coupled to respective ones of the charge terminals T₁ and T₂. In one embodiment, the coil L_(1j) is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the polyphase waveguide probe 200 j. While the electrical length of the coil L_(1j) is specified as approximately one-half (½) the wavelength at the operating frequency, it is understood that the coil L_(1j) may be specified with an electrical length at other values as was discussed with reference to the polyphase waveguide probe 200 a (FIG. 7A) described above. In addition, the probe coupling circuit 209 j includes a tap 223 on the coil L_(1j) that is coupled to the lossy conducting medium 203.

The excitation source 213 is magnetically coupled to the probe coupling circuit 209 j in a manner similar as was mentioned with respect to the polyphase waveguide probe 200 a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil L_(P) that acts as a primary and the coil L_(1j) acts as a secondary. The coil L_(P) can be positioned at any location along the coil L_(1j). Also, the coil L_(P) can be positioned above or below the tap 223.

In order to adjust the polyphase waveguide probe 200 j for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T₁ and T₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T₁ and T₂ may be altered. In addition, the size of the coil L_(1j) may be altered by adding or eliminating turns or by changing some other dimension of the coil L_(1j). Further, the position of the tap 223 on the coil L_(1j) may be adjusted.

With reference to the various embodiments of the polyphase waveguide probes 200 a-j in FIGS. 7A-J, each of the polyphase waveguide probes 200 a-j may be excited to transmit energy conveyed in the form of a guided wave, or waveguide mode along the surface of the lossy conducting medium 203. To facilitate such transmission, the elements of each of the polyphase waveguide probes 200 a-j may be adjusted to impose a desired voltage magnitude and phase on the respective charge terminals T₁ and T₂ when the respective polyphase waveguide probe 200 a-j is excited. Such excitation may occur by applying energy from an excitation source 213 to the respective polyphase waveguide probe 200 a-j as was described above.

The voltage magnitudes and phases imposed on the charge terminals T₁ and T₂ may be adjusted in order to substantially synthesize the fields that are substantially mode-matched to the guided or Zenneck surface-waveguide mode of the lossy conducting medium 203 at the site of transmission given the local permittivity ∈_(r), conductivity a, and potentially other parameters of the lossy conducting medium 203. The waveguide mode of the surface-guided wave is expressed in Equations (21), (22), and (23) set forth above. This surface-waveguide mode has a radial surface current density expressed in Equation (20) in Amperes per meter.

It is understood that it may be difficult to synthesize fields that exactly match the surface-waveguide mode expressed in Equations (21), (22), and (23) set forth above. However, a guided surface wave may be launched if such fields at least approximate the surface-waveguide mode. According to various embodiments, the fields are synthesized to match the surface-waveguide mode within an acceptable engineering tolerance so as to launch a guided surface wave.

Likewise, it may be difficult to synthesize a radial surface current density that exactly matches the radial surface current density of the Zenneck surface-waveguide mode, where the synthesized radial surface current density results from the synthesized fields described above. According to various embodiments, the polyphase waveguide probes 200 may be adjusted to match the radial surface current density of the guided surface-waveguide mode within an acceptable engineering tolerance so as to launch a Zenneck surface wave mode. By creating specific charge distributions plus their images at complex distances, the various polyphase waveguide probes 200 a-j set forth above excite surface currents, the fields of which are designed to approximately match a propagating Zenneck surface wave mode and a Zenneck surface wave is launched. By virtue of this complex image technique inherent in the various polyphase waveguide probes 200 a-j described above, one is able to substantially mode-match to the surface waveguide modes that the guiding interface wants to support at the location of transmission. The guiding interface is the interface between Region 1 (FIG. 2) and Region 2 (FIG. 2) as described above. According to one embodiment, the guiding interface is the interface between the lossy conducting medium 203 presented by the Earth and the atmospheric medium as described above.

When the voltage magnitudes and phases imposed on the charge terminals T₁ and T₂ are adjusted so that they, plus their effective images at complex depths, excite complex surface currents whose fields synthesize the fields that substantially match the Zenneck surface-waveguide mode of the lossy conducting medium 203 at the site of transmission, by virtue of the Leontovich boundary condition, such fields will automatically substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium 203, resulting in zero reflection. This is the condition of wave matching at the boundary.

Referring next to FIGS. 8A, 8B, and 8C, shown are examples of graphs 300 a, 300 b, and 300 c that depict field strength in Volts per meter as a function of distance in kilometers for purposes of comparison between a Zenneck surface wave and conventionally radiated fields. In addition, the various graphs 300 a, 300 b, and 300 c illustrate how the distance of transmission of a Zenneck surface wave varies with the frequency of transmission.

Each graph 300 a, 300 b, and 300 c depicts a corresponding guided field strength curve 303 a, 303 b, and 303 c and corresponding radiated field strength curves 306 a, 306 b, and 306 c. The guided field strength curves 303 a, 303 b, and 303 c were generated assuming various parameters. Specifically, the graphs 300 a, 300 b, and 300 c were calculated with a constant charge Q₁ (FIG. 3) applied to the upper terminal T₁ (FIG. 3) at frequencies of 10 MHz, 1 MHz, and 0.1 MHz, respectively. Constitutive parameters of ∈_(r)=15 and σ=0.008 mhos/m, which are taken from the R-3 map for central Ohio set forth by the Federal Communications Commission (FCC), were assumed for purposes of calculation. The following table provides the assumed polyphase waveguide probe operating parameters for the generation of each of the guided field strength curve 303 a, 303 b, and 303 c.

Height of Self-Cap of Voltage Frequency Terminal Terminal on Term Charge Q₁ Curve (MHz) λ T₁ (m) H/λ T₁ (pF) T₁ (V) (Coulombs) 303a 10 30 0.8 0.027 50 10,000 5 × 10⁻⁷ 303b 1.0 300 8 0.027 100 5,000 5 × 10⁻⁷ 303c 0.1 3000 8 0.0027 100 5,000 5 × 10⁻⁷

In order to have physically realizable operation, the height of terminal T₁ was specified at H_(T1)=8 meters for f=0.1 MHz and 1.0 MHz, but shortened to 0.8 meters at 10 MHz in order to keep the current distribution uniform. Also, the self-capacitance C₁ of the terminal T₁ was set to 100 pF for operation at f=0.1 MHz and 1.0 MHz. This capacitance is unreasonably large for use at 10 MHz, so the self-capacitance C₁ was reduced for that case. However, the resulting terminal charge Q_(T1) which is the controlling parameter for field strength was kept at the same value for all three guided field strength curve 303 a, 303 b, and 303 c.

From the graphs it can be seen that the lower the frequency, the less the propagation attenuation and the fields reach out over greater distances. However, consistent with conservation of energy, the energy density decreases with distance. Another way to state that is that the higher the frequency, the smaller the region over which the energy is spread, so the greater the energy density. Thus, the “knee” of the Zenneck surface wave shrinks in range as the frequency is increased. Alternatively, the lower the frequency, the less the propagation attenuation and the greater the field strength of the Zenneck surface wave at very large distances from the site of transmission using a polyphase waveguide probe 200 (FIG. 6).

The Zenneck surface wave for each case is identified as guided field strength curves 303 a, 303 b, and 303 c, respectively. The Norton ground wave field strength in Volts per meter for a short vertical monopole antenna, of the same height as the respective polyphase waveguide probe 200, with an assumed ground loss of 10 ohms, is represented by the radiated field strength curves 306 a, 306 b, and 306 c, respectively. It is asserted that this is a reasonably realistic assumption for monopole antenna structures operating at these frequencies. The critical point is that a properly mode-matched polyphase waveguide probe launches a guided surface wave which dramatically outperforms the radiation field of any monopole at distances out to just beyond the “knee” in the guided field strength curves 303 a-c of the respective Zenneck surface waves.

Given the foregoing, according to one embodiment, the propagation distance of a guided surface wave varies as a function of the frequency of transmission. Specifically, the lower the transmission frequency, the less the exponential attenuation of the guided surface wave and, therefore, the farther the guided surface wave will propagate. As mentioned above, the field strength of a guided surface wave falls off at a rate of

$\frac{{\mathbb{e}}^{{- \alpha}\; d}}{\sqrt{d}},$ whereas the field strength of a radiated electromagnetic field falls off geometrically, proportional to 1/d, where d is the distance in kilometers. Thus, each of the guided field strength curves 303 a, 303 b, and 303 c feature a knee as was described above. As the frequency of transmission of a polyphase waveguide probe described herein decreases, the knee of the corresponding guided field strength curve 303 a, 303 b, and 303 c will push to the right in the graph.

FIG. 8A shows a guided field strength curve 303 a and a radiated field strength curve 306 a generated at a frequency of 10 Megahertz. As shown, the guided surface wave falls off below 10 kilometers. In FIG. 8B, the guided field strength curve 303 b and the radiated field strength curve 306 b are generated at a frequency of 1 Megahertz. The guided field strength curve 303 b falls off at approximately 100 Kilometers. Finally, in FIG. 8C, the guided field strength curve 303 c and the radiated field strength curve 306 c are generated at a frequency of 100 Kilohertz (which is 0.1 Megahertz). The guided field strength curve 303 c falls off at between 4000-7000 Kilometers.

Note that if the frequency is low enough, it may be possible to transmit a guided surface wave around the entire Earth. It is believed that such frequencies may be at or below approximately 20-25 kilohertz. It should be noted that at such low frequencies, the lossy conducting medium 203 (FIG. 6) ceases to be a plane and becomes a sphere. Thus, when a lossy conducting medium 203 comprises a terrestrial medium, the calculation of guided field strength curves will be altered to take into account the spherical shape at low frequencies where the propagation distances approach size of the terrestrial medium.

Given the foregoing, next some general guidance is provided in constructing a polyphase waveguide probe 200 (FIG. 6) using the terrestrial medium of Earth as the lossy conducting medium 203 according to the various embodiments. As a practical approach, one might specify the frequency of operation and identify the desired field strength of the guided surface wave at a distance of interest from the respective polyphase waveguide probe 200 to be constructed.

Given these parameters, next one may determine the charge Q₁ (FIG. 6) that is to be imposed on the upper charge terminal T₁ (FIG. 6) in order to produce the desired field strength at the specified distance. To determine the charge Q₁ needed, one would need to obtain the permittivity ∈_(r) and the conductivity a of the Earth at the transmission site. These values can be obtained by measurement or by reference to conductivity charts published, for example, by the Federal Communications Commission or the Committee Consultif International Radio (CCIR). When the permittivity ∈_(r), conductivity σ, and the desired field strength at the specified distance are known, the needed charge Q₁ can be determined by direct calculation of the field strength from Zenneck's exact expressions set forth in Equations (21)-(23) above.

Once the needed charge Q₁ is determined, next one would need to identify what self-capacitance C₁ of the charge terminal T₁ at what voltage V would produce the needed charge Q₁ on the charge terminal T₁. A charge Q on any charge terminal T is calculated as Q=CV. In one approach, one can choose what is deemed to be an acceptable voltage V that can be placed on the charge terminal T₁, and then construct the charge terminal T₁ so as to have the required self-capacitance C₁ to achieve the needed charge Q₁. Alternatively, in another approach, one could determine what is an achievable self-capacitance C₁ by virtue of the specific construction of the charge terminal T₁, and then raise the resulting charge terminal T₁ to the required voltage V to achieve the needed charge Q₁.

In addition, there is an issue of operational bandwidth that should be considered when determining the needed self-capacitance C₁ of the charge terminal T₁ and voltage V to be imposed on the charge terminal T₁. Specifically, the bandwidth of the polyphase waveguide probes 200 described herein is relatively large. This results in a significant degree of flexibility in specifying the self-capacitance C₁ or the voltage V as described above. However, it should be understood that as the self-capacitance C₁ is reduced and the voltage V increased, the bandwidth of the resulting polyphase waveguide probe 200 will diminish.

Experimentally, it should be noted that a smaller self-capacitance C₁ may make a given polyphase waveguide probe 200 more sensitive to small variations in the permittivity ∈_(r) or conductivity σ of the Earth at or near the transmission site. Such variation in the permittivity ∈_(r) or conductivity σ might occur due to variation in the climate given the transition between the seasons or due to changes in local weather conditions such as the onset of rain, drought, and/or other changes in local weather. Consequently, according to one embodiment, the charge terminal T₁ may be specified so as to have a relatively large self-capacitance C₁ as is practicable.

Once the self-capacitance C₁ of the charge terminal T₁ and the voltage to be imposed thereon are determined, next the self-capacitance C₂ and physical location of the second charge terminal T₂ are to be determined. As a practical matter, it has been found easiest to specify the self-capacitance C₂ of the charge terminal T₂ to be the same as the self-capacitance C₁ of the charge terminal T₁. This may be accomplished by making the size and shape of the charge terminal T₂ the same as the size and shape of the charge terminal T₁. This would ensure that symmetry is maintained and will avoid the possibility of unusual phase shifts between the two charge terminals T₁ and T₂ that might negatively affect achieving a match with the complex Brewster angle as described above. The fact that the self-capacitances C₁ and C₂ are the same for both charge terminals T₁ and T₂ will result in the same voltage magnitudes on the charge terminals T₁ and T₂. However, it is understood that the self-capacitances C₁ and C₂ may differ, and the shape and size of the charge terminals T₁ and T₂ may differ.

To promote symmetry, the charge terminal T₂ may be positioned directly under the charge terminal T₁ along the vertical axis z (FIG. 6) as described above. Alternatively, it may be possible to position the charge terminal T₂ at some other location with some resulting effect.

The distance between the charge terminals T₁ and T₂ should be specified so as to provide for the best match between the fields generated by the polyphase waveguide probe 200 and the guided surface-waveguide mode at the site of transmission. As a suggested starting point, this distance may be set so that the mutual capacitance C_(M) (FIG. 6) between the charge terminals T₁ and T₂ is the same or less than the isolated capacitance C₁ on the charge terminal T₁. Ultimately, one should specify the distance between the charge terminals T₁ and T₂ to make the mutual capacitance C_(M) as small as is practicable. The mutual capacitance C_(M) may be determined by measurement, and the charge terminals T₁ and T₂ may be positioned accordingly.

Next, the proper height h=H₁−H₂ (FIGS. 7A-J) of the polyphase waveguide probe 200 is determined. The so-called “image complex-depth” phenomenon comes into bearing here. This would entail consideration of the superposed fields on the surface of the Earth from the charge reservoirs T₁ and T₂ that have charges Q₁ and Q₂, and from the subsurface images of the charges Q₁ and Q₂ as height h is varied. Due to the significant number of variables to consider to ensure that a given polyphase waveguide probe 200 is mode matched with the guided surface-waveguide mode of the Earth at the site of transmission, a practical starting point is a height h at which the bound capacitance of each of the charge reservoirs T₁ and T₂ with respect to the ground is negligible such that the capacitance associated with the charge terminals T₁ and T₂ is essentially their isolated self-capacitance C₁ and C₂, respectively.

Another consideration to take into account when determining the height h associated with a polyphase waveguide probe 200 is whether radiation is to be avoided. Specifically, as the height h of the polyphase waveguide probe 200 approaches an appreciable portion of a wavelength at the frequency of operation, the radiation resistance R_(r) will grow quadratically with height h and radiation will begin to dominate over the generation of a guided surface wave as described above. One benchmark set forth above that ensures the Zenneck surface wave will dominate over any radiation is to make sure the height h is less than 10% of the wavelength at the frequency of operation, although other benchmarks may be specified. In some cases, it may be desired to allow some degree of radiation to occur in addition to launching a guided surface wave, where the height h may be specified accordingly.

Next, the probe coupling circuit 209 (FIG. 6) is specified to provide for the voltage phase between the charge terminals T₁ and T₂. The voltage phase appears to have a significant effect on creating fields that mode-match the guided surface-waveguide mode at the site of transmission. Assuming that the placement of the charge terminals T₁ and T₂ is along the vertical z axis to promote symmetry, the probe coupling circuit 209 may be specified to provide for a voltage phase differential of 180 degrees on the charge terminals T₁ and T₂. That is to say, the probe coupling circuit 209 is specified so that voltage V on the charge terminal T₁ is 180 degrees out of phase with respect to the voltage on the charge terminal T₂.

As was described above, one example approach is to place a coil L_(1a) (FIG. 7A) between the charge terminals T₁ and T₂ as described above with reference to the polyphase waveguide probe 200 a and adjust the coil L_(1a) until the resulting system is electrically half-wave resonant. This would place a voltage V on the charge terminal T₁ and voltage −V on the charge terminal T₂ such that the largest voltages are placed on the charge terminals T₁ and T₂ 180 degrees out of phase.

The excitation source 213 (FIG. 6) may then be coupled to the probe coupling circuit 209 and the output voltage adjusted to achieve the required voltage V to provide for the needed charge Q₁ as described above. The excitation source 213 may be coupled via magnetic coupling, capacitive coupling, or conductive coupling (direct) to the probe coupling circuit 209. Note that the output of the excitation source 213 may be stepped up using a transformer or via some other approach if necessary. The location of the coil L_(1a) can be at any location such as down on the ground by the excitation source 213. Alternatively, as per best RF practice, the coil L_(1a) can be positioned directly between the charge reservoirs T₁ and T₂. Principles of impedance matching may be applied when coupling the excitation source 213 to the probe coupling circuit 209.

Note that the phase differential does not necessarily have to be 180 degrees. To this end, one has the option of raising and lowering one or both of the charge terminals T₁ and/or T₂, adjusting the voltages V on the charge terminals T₁ and/or T₂, or adjusting the probe coupling circuit 209 to adjust the voltage magnitudes and phases to create fields that most closely match the guided surface-waveguide mode in order to generate a guided surface wave.

Experimental Results

The above disclosures are supported by experimental measurements and documentation. With reference to FIG. 9, shown is a graph that presents the measured field strength of an electromagnetic field transmitted by one embodiment of an experimental polyphase waveguide probe measured on Oct. 14, 2012 in Plymouth, N.H. The frequency of transmission was 59 MHz with a voltage of 60 mV imposed on the charge terminal T₁ of the experimental polyphase waveguide probe. The self-capacitance C₁ of the experimental polyphase waveguide probe was 8.5 pF. The conductivity σ of the ground at the test site is 0.0002 mhos/m, and the permittivity ∈_(r) of the ground at the test site was 5. These values were measured in situ at the frequency in use.

The graph includes a guided field strength curve 400 that is labeled a “Zenneck” curve at 80% efficiency and a radiated field strength curve 403 that is labeled a “Norton” curve at 100% radiation efficiency, which is the best possible. To this end, the radiated field strength curve 403 represents the radiated electromagnetic fields that would be generated by a ¼ wavelength monopole antenna operating at a frequency of 59 MHz. The circles 406 on the graph represent measured field strengths produced by the experimental polyphase waveguide probe. The field strength measurements were performed with a NIST-traceable Potomac Instruments FIM-71 commercial VHF field strength meter. As can be seen, the measured field strengths fall along the theoretical guided field strength curve 400. These measured field strengths are consistent with the propagation of a guided or Zenneck surface wave.

Referring next to FIG. 10, shown is a graph that presents the measured phase of the transmitted electromagnetic wave from the experimental polyphase waveguide probe. The curve J(r) indicates the phase of the fields incident to the currents J₁ and J₂ with a transition between the currents J₁ and J₂ as shown. The curve 503 indicates the asymptote depicting the phase of the current J₁, and the curve 506 indicates the asymptote depicting the phase of the current J₂. A difference of approximately 45 degrees generally exists between the phases of the respective currents J₁ and J₂. The circles 509 indicate measurements of the phase of the current J(r) generated by the experimental polyphase waveguide probe operating at 59 MHz as with FIG. 9. As shown, the circles 509 fall along the curve J(r) indicating that there is a transition of the phase of the current J(r) from the curve 503 to the curve 506. This indicates that the phase of the current J(r) generated by the experimental polyphase waveguide probe transitions from the phase generated by the close-in current J₁ to the far-out current J₂. Thus, these phase measurements are consistent with the phase with the presence of a guided or Zenneck surface wave.

With reference to FIG. 11 shown is a graph of a second set of measured data that depicts the field strength of an electromagnetic field transmitted by a second embodiment of an experimental polyphase waveguide probe measured on Nov. 1, 2003 in the vicinity of Ashland, N.H. and across the region north of Lake Winnipesaukee. The frequency of transmission was 1850 kHz with a voltage of 1250 V imposed on the charge terminal T₁ of the experimental polyphase waveguide probe. The experimental polyphase waveguide probe had a physical height of H₁=2 meters. The self-capacitance C₁ of the experimental polyphase waveguide probe in this experiment, which was a flat conducting disk of 1 meter radius, was measured to be 70 pF. The polyphase waveguide probe was arranged as illustrated in FIG. 7J, with spacing h=1 meter and the height of the charge terminal T₂ above the ground (the lossy conducting medium 203) being H₂=1 meter. The average conductivity a of the ground in the vicinity of experimentation was 0.006 mhos/m, and the relative permittivity ∈_(r) of the ground was on the order of 15. These were determined at the frequency in use.

The graph includes a guided field strength curve 600 that is launched by the experimental polyphase waveguide probe, labeled as “Zenneck” curve at 85% efficiency, and a radiated field strength curve 603 that is labeled a “Norton” curve as radiated from a resonated monopole of the same height, H₂=2 meters, over a ground screen composed of 20 radial wires equally spaced and of length 200 feet each. To this end, the radiated field strength curve 603 represents the conventional Norton ground wave field radiated from a conventional stub monopole antenna operating at a frequency of 1850 kHz over the lossy Earth. The circles 606 on the graph represent measured field strengths produced by the experimental polyphase waveguide probe.

As can be seen, the measured field strengths fall closely along the theoretical Zenneck guided field strength curve 600. Special mention of the field strength measured at the r=7 mile point may be made. This field strength data point was measured adjacent to the shore of a lake, and this may account for the data departing slightly above the theoretical Zenneck guided field strength curve 600, i.e. the constitutive parameters, ∈_(r) and/or σ, at that location are likely to have departed significantly from the path-average constitutive parameters.

The field strength measurements were performed with a NIST-traceable Potomac Instruments FIM-41 MF/HF field strength meter. The measured field strength data are consistent with the presence of a guided or Zenneck surface wave. It is apparent from the experimental data that the measured field strengths observed at distances less than 15 miles could not possibly be due to conventional Norton ground wave propagation, and can only be due to guided surface wave propagation launched by the polyphase probe operating as disclosed above. Under the given 1.85 MHz experimental conditions, out at 20 miles it appears that a Norton ground wave component has finally overtaken the Zenneck surface wave component.

A comparison of the measured Zenneck surface wave data shown in FIG. 9 at 59 MHz with the measured data in FIG. 11 at 1.85 MHz illustrates the great advantage of employing a polyphase waveguide probe according to the various embodiments at lower frequencies.

These experimental data confirm that the present polyphase waveguide probes, comprising a plurality of appropriately phased and adjusted charge terminals, as taught herein, induce a phase-advanced surface current with a unique phase boost of arg(√{square root over (γ)}), and whose fields synthesize surface illumination at the complex Brewster angle for the lossy boundary as disclosed herein. The consequence is the efficient launching of cylindrical Zenneck-like wave propagation, guided by the boundary surface as an evanescent, single-conductor radial transmission-line mode, which attenuates as not as

$\frac{{\mathbb{e}}^{{- \alpha}\; d}}{\sqrt{d}},$ not as a radiation field, which would decrease as 1/d due to geometrical spreading.

Referring next to FIGS. 12A, 12B, and 13, shown are examples of generalized receive circuits for using the surface-guided waves in wireless power delivery systems. FIGS. 12A and 12B include a linear probe 703 and a tuned resonator 706. FIG. 13 is a magnetic coil 709 according to various embodiments of the present disclosure. According to various embodiments, each one of the linear probe 703, the tuned resonator 706, and the magnetic coil 709 may be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium 203 (FIG. 6) according to various embodiments. As mentioned above, in one embodiment the lossy conducting medium 203 comprises a terrestrial medium.

With specific reference to FIG. 12A, the open-circuit terminal voltage at the output terminals 713 of the linear probe 703 depends upon the effective height of the linear probe 703. To this end, the terminal point voltage may be calculated as V _(T)=∫₀ ^(h) ^(e) E _(inc) ·dl,  (43) where E_(inc) is the strength of the electric field in vector on the linear probe 703 in Volts per meter, dl is an element of integration along the direction of the linear probe 703, and h_(e) is the effective height of the linear probe 703. An electrical load 716 is coupled to the output terminals 713 through an impedance matching network 719.

When the linear probe 703 is subjected to a guided surface wave as described above, a voltage is developed across the output terminals 713 that may be applied to the electrical load 716 through a conjugate impedance matching network 719 as the case may be. In order to facilitate the flow of power to the electrical load 716, the electrical load 716 should be substantially impedance matched to the linear probe 703 as will be described below.

Referring to FIG. 12B, the tuned resonator 706 includes a charge terminal T_(R) that is elevated above the lossy conducting medium 203. The charge terminal T_(R) has a self-capacitance C_(R). In addition, there may also be a bound capacitance (not shown) between the charge terminal T_(R) and the lossy conducting medium 203 depending on the height of the charge terminal T_(R) above the lossy conducting medium 203. The bound capacitance should preferably be minimized as much as is practicable, although this may not be entirely necessary in every instance of a polyphase waveguide probe 200.

The tuned resonator 706 also includes a coil L_(R). One end of the coil L_(R) is coupled to the charge terminal T_(R), and the other end of the coil L_(R) is coupled to the lossy conducting medium 203. To this end, the tuned resonator 706 (which may also be referred to as tuned resonator L_(R)-C_(R)) comprises a series-tuned resonator as the charge terminal C_(R) and the coil L_(R) are situated in series. The tuned resonator 706 is tuned by adjusting the size and/or height of the charge terminal T_(R), and/or adjusting the size of the coil L_(R) so that the reactive impedance of the structure is substantially eliminated.

For example, the reactance presented by the self-capacitance C_(R) is calculated as

$\frac{1}{j\;\omega\; C_{R}}.$ Note that the total capacitance of the tuned resonator 706 may also include capacitance between the charge terminal T_(R) and the lossy conducting medium 203, where the total capacitance of the tuned resonator 706 may be calculated from both the self-capacitance C_(R) and any bound capacitance as can be appreciated. According to one embodiment, the charge terminal T_(R) may be raised to a height so as to substantially reduce or eliminate any bound capacitance. The existence of a bound capacitance may be determined from capacitance measurements between the charge terminal T_(R) and the lossy conducting medium 203.

The inductive reactance presented by a discrete-element coil L_(R) may be calculated as jωL, where L is the lumped-element inductance of the coil L_(R). If the coil L_(R) is a distributed element, its equivalent terminal-point inductive reactance may be determined by conventional approaches. To tune the tuned resonator 706, one would make adjustments so that the inductive reactance presented by the coil L_(R) equals the capacitive reactance presented by the tuned resonator 706 so that the resulting net reactance of the tuned resonator 706 is substantially zero at the frequency of operation. An impedance matching network 723 may be inserted between the probe terminals 721 and the electrical load 726 in order to effect a conjugate-match condition for maxim power transfer to the electrical load 726.

When placed in the presence of a guided surface wave, generated at the frequency of the tuned resonator 706 and the conjugate matching network 723, as described above, maximum power will be delivered from the surface guided wave to the electrical load 726. That is, once conjugate impedance matching is established between the tuned resonator 706 and the electrical load 726, power will be delivered from the structure to the electrical load 726. To this end, an electrical load 726 may be coupled to the tuned resonator 706 by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling. The elements of the coupling network may be lumped components or distributed elements as can be appreciated. In the embodiment shown in FIG. 12B, magnetic coupling is employed where a coil L_(S) is positioned as a secondary relative to the coil L_(R) that acts as a transformer primary. The coil L_(S) may be link coupled to the coil L_(R) by geometrically winding it around the same core structure and adjusting the coupled magnetic flux as can be appreciated. In addition, while the tuned resonator 706 comprises a series-tuned resonator, a parallel-tuned resonator or even a distributed-element resonator may also be used.

Referring to FIG. 13, the magnetic coil 709 comprises a receive circuit that is coupled through an impedance coupling network 733 to an electrical load 736. In order to facilitate reception and/or extraction of electrical power from a guided surface wave, the magnetic coil 709 may be positioned so that the magnetic flux of the guided surface wave, H_(φ), passes through the magnetic coil 709, thereby inducing a current in the magnetic coil 709 and producing a terminal point voltage at its output terminals 729. The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by Ψ=∫∫_(A) _(CS) μ_(r)μ_(o)

·{circumflex over (n)}dA  (44) where ψ is the coupled magnetic flux, μ_(r) is the effective relative permeability of the core of the magnetic coil 709, μ_(o) is the permeability of free space, H is the incident magnetic field strength vector, n is a unit vector normal to the cross-sectional area of the turns, and A_(CS) is the area enclosed by each loop. For an N-turn magnetic coil 709 oriented for maximum coupling to an incident magnetic field that is uniform over the cross-sectional area of the magnetic coil 709, the open-circuit induced voltage appearing at the output terminals 729 of the magnetic coil 709 is

$\begin{matrix} {{V = {{{- N}\frac{\mathbb{d}\Psi}{\mathbb{d}t}} \approx {{- {j\omega\mu}_{r}}\mu_{0}H\; A_{CS}}}},} & (45) \end{matrix}$ where the variables are defined above. The magnetic coil 709 may be tuned to the guided wave frequency either as a distributed resonator or with an external capacitor across its output terminals 729, as the case may be, and then impedance-matched to an external electrical load 736 through a conjugate impedance matching network 733.

Assuming that the resulting circuit presented by the magnetic coil 709 and the electrical load 736 are properly adjusted and conjugate impedance matched, via impedance matching network 733, then the current induced in the magnetic coil 709 may be employed to optimally power the electrical load 736. The receive circuit presented by the magnetic coil 709 provides an advantage in that it does not have to be physically connected to the ground.

With reference to FIGS. 12A, 12B, and 13, the receive circuits presented by the linear probe 703, the tuned resonator 706, and the magnetic coil 709 each facilitate receiving electrical power transmitted from any one of the embodiments of polyphase waveguide probes 200 described above. To this end, the energy received may be used to supply power to an electrical load 716/726/736 via a conjugate matching network as can be appreciated. This contrasts with the signals that may be received in a receiver that were transmitted in the form of a radiated electromagnetic field. Such signals have very low available power and receivers of such signals do not load the transmitters.

It is also characteristic of the present guided surface waves generated using the polyphase waveguide probes 200 described above that the receive circuits presented by the linear probe 703, the tuned resonator 706, and the magnetic coil 709 will load the excitation source 213 (FIG. 3) that is applied to the polyphase waveguide probe 200, thereby generating the guided surface wave to which such receive circuits are subjected. This reflects the fact that the guided surface wave generated by a given polyphase waveguide probe 200 described above comprises a transmission line mode. By way of contrast, a power source that drives a radiating antenna that generates a radiated electromagnetic wave is not loaded by the receivers, regardless of the number of receivers employed.

Thus, together a given polyphase waveguide probe 200 and receive circuits in the form of the linear probe 703, the tuned resonator 706, and/or the magnetic coil 709 can together make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a polyphase waveguide probe 200 as set forth above depends upon the frequency, it is possible that wireless power distribution can be achieved across wide areas and even globally.

The conventional wireless-power transmission/distribution systems extensively investigated today include “energy harvesting” from radiation fields and also sensor coupling to inductive or reactive near-fields. In contrast, the present wireless-power system does not waste power in the form of radiation which, if not intercepted, is lost forever. Nor is the presently disclosed wireless-power system limited to extremely short ranges as with conventional mutual-reactance coupled near-field systems. The wireless-power system disclosed herein probe-couples to the novel surface-guided transmission line mode, which is equivalent to delivering power to a load by a wave-guide or a load directly wired to the distant power generator. Not counting the power required to maintain transmission field strength plus that dissipated in the surface waveguide, which at extremely low frequencies is insignificant relative to the transmission losses in conventional high-tension power lines at 60 Hz, all the generator power goes only to the desired electrical load. When the electrical load demand is terminated, the source power generation is relatively idle.

Referring next to FIG. 14A shown is a schematic that represents the linear probe 703 and the tuned resonator 706. FIG. 14B shows a schematic that represents the magnetic coil 709. The linear probe 703 and the tuned resonator 706 may each be considered a Thevenin equivalent represented by an open-circuit terminal voltage source V_(S) and a dead network terminal point impedance Z_(S). The magnetic coil 709 may be viewed as a Norton equivalent represented by a short-circuit terminal current source I_(S) and a dead network terminal point impedance Z_(S). Each electrical load 716/726/736 (FIGS. 12A-B and FIG. 13) may be represented by a load impedance Z_(L). The source impedance Z_(S) comprises both real and imaginary components and takes the form Z_(S)=R_(S)+jX_(S).

According to one embodiment, the electrical load 716/726/736 is impedance matched to each receive circuit, respectively. Specifically, each electrical load 716/726/736 presents through a respective impedance matching network 719/723/733 a load on the probe network specified as Z_(L)′ expressed as Z_(L)′=R_(L)′+j X_(L)′, which will be equal to Z_(L)′=Z_(s)*=R_(S)−j X_(S), where the presented load impedance Z_(L)′ is the complex conjugate of the actual source impedance Z_(S). The conjugate match theorem, which states that if, in a cascaded network, a conjugate match occurs at any terminal pair then it will occur at all terminal pairs, then asserts that the actual electrical load 716/726/736 will also see a conjugate match to its impedance, Z_(L)′. See Everitt, W. L. and G. E. Tanner, Communication Engineering, McGraw-Hill, 3^(rd) edition, 1956, p. 407. This ensures that the respective electrical load 716/726/736 is impedance matched to the respective receive circuit and that maximum power transfer is established to the respective electrical load 716/726/736.

It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. In addition, all optional and preferred features and modifications of the described embodiments and dependent claims are usable in all aspects of the disclosure taught herein. Furthermore, the individual features of the dependent claims, as well as all optional and preferred features and modifications of the described embodiments are combinable and interchangeable with one another. 

The invention claimed is:
 1. A method, comprising: positioning a receive circuit relative to a terrestrial medium; and receiving, via the receive circuit, energy conveyed in a form of a Zenneck surface wave propagating on a surface of the terrestrial medium in a guided transmission line mode, the Zenneck surface wave launched by electromagnetic fields generated by a polyphase waveguide probe, where the electromagnetic fields are substantially mode-matched to a guided surface wave mode of the terrestrial medium, wherein the guided surface wave mode is substantially expressed as $\begin{matrix} {{H_{\phi} = {\frac{{- \gamma}\; I_{o}}{4}{\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- {j\gamma}}\;\rho} \right)}}},} \\ {{E_{p} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{u_{2}}{j\;{\omega ɛ}_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},{and}} \\ {E_{z} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{- \gamma}{{\omega ɛ}_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}} \end{matrix}$ where H_(φ) is an azimuthal magnetic field strength, E_(ρ) is a radial electric field strength, E_(z) is a vertical electric field strength, where γ is a surface wave radial propagation constant given by γ=j√{square root over (k_(o) ²+u₂ ²)} and u₂ is a vertical propagation constant given by ${u_{2} = \frac{{- j}\; k_{o}}{\sqrt{1 + \left( {ɛ_{r} - {j\; x}} \right)}}},{{{where}\mspace{14mu} x} \equiv \frac{\sigma}{{\omega ɛ}_{o}}},$ σ is a conductivity of the terrestrial medium, ω is equal to 2πf, where f is a frequency of excitation of the polyphase waveguide probe, ∈_(o) is a permittivity of free space, ∈_(r) is a relative permittivity of the terrestrial medium, and a free-space wave number k_(o) is equal to $\frac{2\pi}{\lambda_{o}},$ where λ_(o) is a free-space wavelength of the polyphase waveguide probe, j is equal to √{square root over (−1)}, ρ is a radial coordinate, z is a vertical coordinate normal to the terrestrial medium, φ is an azimuthal coordinate, I_(o) is a net polyphase probe current, H₁ ⁽²⁾(−jγρ) is a Hankel function of a second kind and first order with complex argument −jγρ, and H₀ ⁽²⁾(−jγρ) is a Hankel function of a second kind and zero order with complex argument −jγρ, for an e^(+jωt) time variation, where t is time.
 2. The method of claim 1, wherein the receive circuit loads an excitation source coupled to the polyphase waveguide probe that generates electromagnetic fields which synthesize a wave front incident at a complex Brewster angle of the terrestrial medium surrounding the polyphase waveguide probe thereby launching the Zenneck surface wave.
 3. The method of claim 1, wherein the energy comprises electrical power, and the method further comprises applying the electrical power to an electrical load coupled to the receive circuit, where the electrical power is used as a power source for the electrical load.
 4. The method of claim 1, comprising impedance matching an electrical load to the receive circuit.
 5. The method of claim 4, further comprising establishing a maximum power transfer from the receive circuit to the electrical load.
 6. The method of claim 1, wherein the receive circuit comprises a magnetic coil positioned to couple with electromagnetic fields of the Zenneck surface wave.
 7. The method of claim 1, wherein the receive circuit comprises a linear probe positioned to couple with electromagnetic fields of the Zenneck surface wave.
 8. The method of claim 1, wherein the receive circuit comprises a tuned resonator coupled to the terrestrial medium, the tuned resonator positioned to couple with electromagnetic fields of the Zenneck surface wave.
 9. An apparatus, comprising: a receive circuit that receives energy conveyed in a form of a Zenneck surface wave propagating along a surface of a terrestrial medium in a guided transmission line mode, the Zenneck surface wave launched by electromagnetic fields generated by a polyphase waveguide probe, where the electromagnetic fields are substantially mode-matched to a guided surface wave mode of the terrestrial medium, wherein the guided surface wave mode is substantially expressed as $\begin{matrix} {{H_{\phi} = {\frac{{- \gamma}\; I_{o}}{4}{\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},} \\ {{E_{p} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{u_{2}}{j\;{\omega ɛ}_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},{and}} \\ {E_{z} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{- \gamma}{{\omega ɛ}_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}} \end{matrix}$ where H_(φ) is an azimuthal magnetic field strength, E_(ρ) is a radial electric field strength, E_(z) is a vertical electric field strength, where γ is a surface wave radial propagation constant given by γ=j√{square root over (k_(o) ²+u₂ ²)} and u₂ is a vertical propagation constant given by ${u_{2} = \frac{{- j}\; k_{o}}{\sqrt{1 + \left( {ɛ_{r} - {j\; x}} \right)}}},{{{where}\mspace{14mu} x} \equiv \frac{\sigma}{{\omega ɛ}_{o}}},$ σ is a conductivity of the terrestrial medium, ω is equal to 2πf, where f is a frequency of excitation of the polyphase waveguide probe, ∈_(o) is a permittivity of free space, ∈_(r) is a relative permittivity of the terrestrial medium, and a free-space wave number k_(o) is equal to $\frac{2\pi}{\lambda_{o}},$ where λ_(o) is a free-space wavelength of the polyphase waveguide probe, j is equal to √{square root over (−1)}, ρ is a radial coordinate, z is a vertical coordinate normal to the terrestrial medium, φ is an azimuthal coordinate, I_(o) is a net polyphase probe current, H₁ ⁽²⁾(−jγρ) is a Hankel function of a second kind and first order with complex argument −jγρ, and H₀ ⁽²⁾(−jγρ) is a Hankel function of a second kind and zero order with complex argument −jγρ, for an e^(+jωt) time variation, where t is time.
 10. The apparatus of claim 9, wherein the receive circuit is configured to load an excitation source coupled to the polyphase waveguide probe that generates electromagnetic fields which synthesize a wave front incident at a complex Brewster angle of the terrestrial medium surrounding the polyphase waveguide probe thereby launching the Zenneck surface wave.
 11. The apparatus of claim 9, wherein the energy comprises electrical power, and the receive circuit is coupled to an electrical load, and wherein the electrical power is applied to the electrical load, the electrical power being employed as a power source for the electrical load.
 12. The apparatus of claim 11, wherein the electrical load is impedance matched with the receive circuit.
 13. The apparatus of claim 9, wherein the receive circuit comprises a magnetic coil that couples with electromagnetic fields of the Zenneck surface wave.
 14. The apparatus of claim 9, wherein the receive circuit comprises a linear probe that couples with electromagnetic fields of the Zenneck surface wave.
 15. The apparatus of claim 9, wherein the receive circuit comprises a tuned resonator that couples with electromagnetic fields of the Zenneck surface wave.
 16. The apparatus of claim 15, wherein the tuned resonator comprises a series tuned resonator.
 17. The apparatus of claim 15, wherein the tuned resonator comprises a parallel tuned resonator.
 18. The apparatus of claim 15, wherein the tuned resonator comprises a distributed tuned resonator.
 19. A power transmission system, comprising: a polyphase waveguide probe that transmits electrical energy in a form of a guided surface wave along a surface of a terrestrial medium, where the polyphase waveguide probe generates electromagnetic fields that are substantially mode-matched to a guided surface wave mode of the terrestrial medium, wherein the guided surface wave mode is substantially expressed as $\begin{matrix} {{H_{\phi} = {\frac{{- \gamma}\; I_{o}}{4}{\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},} \\ {{E_{p} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{u_{2}}{j\;{\omega ɛ}_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}},{and}} \\ {E_{z} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{- \gamma}{{\omega ɛ}_{o}} \right){\mathbb{e}}^{{- u_{2}}z}{H_{0}^{(2)}\left( {{- j}\;\gamma\;\rho} \right)}}} \end{matrix}$ where H_(φ) is an azimuthal magnetic field strength, E_(ρ) is a radial electric field strength, E_(z) is a vertical electric field strength, where γ is a surface wave radial propagation constant given by γ=j√{square root over (k_(o) ²+u₂ ²)} and u₂ is a vertical propagation constant given by ${u_{2} = \frac{{- j}\; k_{o}}{\sqrt{1 + \left( {ɛ_{r} - {j\; x}} \right)}}},{{{where}\mspace{14mu} x} \equiv \frac{\sigma}{{\omega ɛ}_{o}}},$ σ is a conductivity of the terrestrial medium, ω is equal to 2πf, where f is a frequency of excitation of the polyphase waveguide probe, ∈_(o) is a permittivity of free space, ∈_(r) is a relative permittivity of the terrestrial medium, and a free-space wave number k_(o) is equal to $\frac{2\pi}{\lambda_{o}},$ where λ_(o) is a free-space wavelength of the polyphase waveguide probe, j is equal to √{square root over (−1)}, ρ is a radial coordinate, z is a vertical coordinate normal to the terrestrial medium, φ is an azimuthal coordinate, I_(o) is a net polyphase probe current, H₁ ⁽²⁾(−jγρ) is a Hankel function of a second kind and first order with complex argument −jγρ, and H₀ ⁽²⁾(−jγρ) is a Hankel function of a second kind and zero order with complex argument −jγρ, for an e^(+jωt) time variation, where t is time; and a receive circuit that receives the electrical energy transmitted through the guided surface wave.
 20. The power transmission system of claim 19, wherein the receive circuit loads the polyphase waveguide probe.
 21. The power transmission system of claim 19, wherein an electrical load is coupled to the receive circuit and the electrical energy is used as a power source for the electrical load.
 22. The power transmission system of claim 21, wherein the electrical load is impedance matched to the receive circuit.
 23. The power transmission system of claim 21, wherein a maximum power transfer is established from the receive circuit to the electrical load. 